From 2e499b2e7d106ad24703268dca0f6d340fcb5e8b Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 10:43:47 +0200
Subject: [PATCH 01/37] action on equivalences of functions

---
 .../delooping-sign-homomorphism.lagda.md      |   4 +-
 src/foundation.lagda.md                       |   1 +
 .../action-on-equivalences-functions.lagda.md | 137 ++++++++++++++++++
 ...tion-on-identifications-functions.lagda.md |   4 +-
 src/foundation/univalence.lagda.md            |  10 +-
 src/group-theory/loop-groups-sets.lagda.md    |   4 +-
 6 files changed, 149 insertions(+), 11 deletions(-)
 create mode 100644 src/foundation/action-on-equivalences-functions.lagda.md

diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index 6e0e2e613d..951248c1cb 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -1098,7 +1098,7 @@ module _
             ( λ r → eq-pair-Σ r (eq-is-prop is-prop-type-trunc-Prop))
             ( ap inv
               ( inv
-                ( compute-eq-equiv
+                ( compute-eq-equiv-equiv-comp
                   ( raise l4 (Fin 2))
                   ( equivalence-class (R (n +ℕ 2) (Fin-UU-Fin l1 (n +ℕ 2))))
                   ( raise l4 (Fin 2))
@@ -1113,7 +1113,7 @@ module _
                 ( ap
                   ( _∙ eq-counting-equivalence-class-R n)
                   ( inv
-                    ( compute-eq-equiv
+                    ( compute-eq-equiv-equiv-comp
                       ( raise l4 (Fin 2))
                       ( equivalence-class (R (n +ℕ 2) (Fin-UU-Fin l1 (n +ℕ 2))))
                       ( equivalence-class (R (n +ℕ 2) (Fin-UU-Fin l1 (n +ℕ 2))))
diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index c251b52600..0e4c08a344 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -10,6 +10,7 @@ open import foundation.0-images-of-maps public
 open import foundation.0-maps public
 open import foundation.1-types public
 open import foundation.2-types public
+open import foundation.action-on-equivalences-functions public
 open import foundation.action-on-identifications-binary-functions public
 open import foundation.action-on-identifications-dependent-functions public
 open import foundation.action-on-identifications-functions public
diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
new file mode 100644
index 0000000000..336fcbe626
--- /dev/null
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -0,0 +1,137 @@
+# Action on equivalences of functions
+
+```agda
+module foundation.action-on-equivalences-functions where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.univalence
+open import foundation.action-on-identifications-functions
+open import foundation.dependent-pair-types
+open import foundation.equality-dependent-function-types
+open import foundation.equivalences
+open import foundation.function-extensionality
+open import foundation.fundamental-theorem-of-identity-types
+open import foundation.universe-levels
+
+open import foundation-core.contractible-types
+open import foundation-core.constant-maps
+open import foundation-core.function-types
+open import foundation-core.functoriality-dependent-pair-types
+open import foundation-core.homotopies
+open import foundation.transport
+open import foundation-core.identity-types
+open import foundation-core.injective-maps
+```
+
+</details>
+
+## Definition
+
+```agda
+ap-eq-equiv :
+  {l1 l2 : Level} {A : UU l2} (B : UU l1 → A) {X Y : UU l1} →
+  X ≃ Y → B X ＝ B Y
+ap-eq-equiv B {X} {Y} e = ap B (eq-equiv X Y e)
+
+ap-equiv :
+  {l1 l2 : Level} (B : UU l1 → UU l2) {X Y : UU l1} →
+  X ≃ Y → B X ≃ B Y
+ap-equiv B = equiv-eq ∘ ap-eq-equiv B
+```
+
+## Properties
+
+### The identity function acts trivially on equivalences
+
+```agda
+ap-equiv-id :
+  {l : Level} {X Y : UU l} (e : X ≃ Y) → (ap-equiv id e) ＝ e
+ap-equiv-id {l} {X} {Y} e =
+  (ap equiv-eq (ap-id (eq-equiv X Y e))) ∙ (is-section-eq-equiv e)
+```
+
+### The action on equivalences of a composite function is the composite of the actions
+
+```agda
+ap-eq-equiv-comp :
+  {l1 l2 l3 : Level} {A : UU l3} (g : UU l2 → A) (f : UU l1 → UU l2)
+  {X Y : UU l1} → ap-eq-equiv (g ∘ f) ~ (ap-eq-equiv g ∘ ap-equiv f)
+ap-eq-equiv-comp g f {X} {Y} e =
+  ( ap-comp g f (eq-equiv X Y e)) ∙
+  ( ap (ap g) (inv (is-retraction-eq-equiv (ap-eq-equiv f e))))
+
+ap-equiv-comp :
+  {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2)
+  {X Y : UU l1} → ap-equiv (g ∘ f) ~ (ap-equiv g ∘ ap-equiv f)
+ap-equiv-comp g f {X} {Y} e =
+  ap equiv-eq (ap-eq-equiv-comp g f {X} {Y} e)
+```
+
+### The action on equivalences of any map preserves `id-equiv`
+
+```agda
+ap-eq-equiv-id-equiv :
+  {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) (A : UU l1) →
+  (ap-eq-equiv f (id-equiv {A = A})) ＝ refl
+ap-eq-equiv-id-equiv f A = ap (ap f) (compute-eq-equiv-id-equiv A)
+
+ap-equiv-id-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) (A : UU l1) →
+  (ap-equiv f (id-equiv {A = A})) ＝ id-equiv {A = f A}
+ap-equiv-id-equiv f A = ap equiv-eq (ap-eq-equiv-id-equiv f A)
+```
+
+### The action on equivalences of any map preserves composition of equivalences
+
+```agda
+ap-eq-equiv-equiv-comp :
+  {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y Z : UU l1} →
+  (e : X ≃ Y) (e' : Y ≃ Z) →
+  ap-eq-equiv f (e' ∘e e) ＝ (ap-eq-equiv f e ∙ ap-eq-equiv f e')
+ap-eq-equiv-equiv-comp f {X} {Y} {Z} e e' =
+    ( ap (ap f) (inv (compute-eq-equiv-equiv-comp X Y Z e e'))) ∙
+    ( ap-concat f (eq-equiv X Y e) (eq-equiv Y Z e'))
+
+ap-equiv-equiv-comp :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X Y Z : UU l1} →
+  (e : X ≃ Y) (e' : Y ≃ Z) →
+  ap-equiv f (e' ∘e e) ＝ (ap-equiv f e' ∘e ap-equiv f e)
+ap-equiv-equiv-comp f {X} {Y} {Z} e e' =
+  ( ap equiv-eq (ap-eq-equiv-equiv-comp f e e')) ∙
+  ( inv (compute-equiv-eq-concat (ap-eq-equiv f e) (ap-eq-equiv f e')))
+```
+
+### The action on equivalences of any map preserves inverses
+
+```agda
+ap-eq-equiv-inv :
+  {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y : UU l1}
+  (e : X ≃ Y) → ap-eq-equiv f (inv-equiv e) ＝ inv (ap-eq-equiv f e)
+ap-eq-equiv-inv f {X} {Y} e =
+  ( ap (ap f) (inv (commutativity-inv-eq-equiv X Y e))) ∙
+  ( ap-inv f (eq-equiv X Y e))
+
+ap-equiv-inv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1}
+  (e : X ≃ Y) → ap-equiv f (inv-equiv e) ＝ inv-equiv (ap-equiv f e)
+ap-equiv-inv f {X} {Y} e =
+  ( ap equiv-eq (ap-eq-equiv-inv f e)) ∙
+  ( inv (commutativity-inv-equiv-eq (f X) (f Y) (ap-eq-equiv f e)))
+```
+
+### The action on equivalences of a constant map is constant
+
+```agda
+ap-eq-equiv-const :
+  {l1 l2 : Level} {B : UU l2} (b : B) {X Y : UU l1}
+  (e : X ≃ Y) → (ap-eq-equiv (const (UU l1) B b) e) ＝ refl
+ap-eq-equiv-const b {X} {Y} e = ap-const b (eq-equiv X Y e)
+
+ap-equiv-const :
+  {l1 l2 : Level} (B : UU l2) {X Y : UU l1}
+  (e : X ≃ Y) → (ap-equiv (const (UU l1) (UU l2) B) e) ＝ id-equiv
+ap-equiv-const B {X} {Y} e = ap equiv-eq (ap-eq-equiv-const B e)
+```
diff --git a/src/foundation/action-on-identifications-functions.lagda.md b/src/foundation/action-on-identifications-functions.lagda.md
index a270aa3f96..7ff1b0867d 100644
--- a/src/foundation/action-on-identifications-functions.lagda.md
+++ b/src/foundation/action-on-identifications-functions.lagda.md
@@ -46,7 +46,7 @@ ap-id :
 ap-id refl = refl
 ```
 
-### The action on identifications of a composite function is the composite of the actions on identifications
+### The action on identifications of a composite function is the composite of the actions
 
 ```agda
 ap-comp :
@@ -84,7 +84,7 @@ ap-concat-eq f p q .(p ∙ q) refl = ap-concat f p q
 ```agda
 ap-inv :
   {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) {x y : A}
-  (p : x ＝ y) → (ap f (inv p)) ＝ (inv (ap f p))
+  (p : x ＝ y) → ap f (inv p) ＝ inv (ap f p)
 ap-inv f refl = refl
 ```
 
diff --git a/src/foundation/univalence.lagda.md b/src/foundation/univalence.lagda.md
index bc0f134384..2d2e60109c 100644
--- a/src/foundation/univalence.lagda.md
+++ b/src/foundation/univalence.lagda.md
@@ -141,18 +141,18 @@ eq-equiv-fam {B = B} {C} = map-inv-is-equiv (is-equiv-equiv-eq-fam B C)
 ### Computations with univalence
 
 ```agda
-compute-equiv-eq :
+compute-equiv-eq-concat :
   {l : Level} {A B C : UU l} (p : A ＝ B) (q : B ＝ C) →
   ((equiv-eq q) ∘e (equiv-eq p)) ＝ equiv-eq (p ∙ q)
-compute-equiv-eq refl refl = eq-equiv-eq-map-equiv refl
+compute-equiv-eq-concat refl refl = eq-equiv-eq-map-equiv refl
 
-compute-eq-equiv :
+compute-eq-equiv-equiv-comp :
   {l : Level} (A B C : UU l) (f : A ≃ B) (g : B ≃ C) →
   ((eq-equiv A B f) ∙ (eq-equiv B C g)) ＝ eq-equiv A C (g ∘e f)
-compute-eq-equiv A B C f g =
+compute-eq-equiv-equiv-comp A B C f g =
   is-injective-map-equiv
     ( equiv-univalence)
-    ( ( inv ( compute-equiv-eq (eq-equiv A B f) (eq-equiv B C g))) ∙
+    ( ( inv ( compute-equiv-eq-concat (eq-equiv A B f) (eq-equiv B C g))) ∙
       ( ( ap
         ( λ e → (map-equiv e g) ∘e (equiv-eq (eq-equiv A B f)))
         ( right-inverse-law-equiv equiv-univalence)) ∙
diff --git a/src/group-theory/loop-groups-sets.lagda.md b/src/group-theory/loop-groups-sets.lagda.md
index 88d3ea96c6..cb6109b443 100644
--- a/src/group-theory/loop-groups-sets.lagda.md
+++ b/src/group-theory/loop-groups-sets.lagda.md
@@ -120,7 +120,7 @@ module _
     map-hom-symmetric-group-loop-group-Set X X
   pr2 hom-symmetric-group-loop-group-Set p q =
     ( ap equiv-eq (distributive-inv-concat p q)) ∙
-      ( inv (compute-equiv-eq (inv q) (inv p)))
+    ( inv (compute-equiv-eq-concat (inv q) (inv p)))
 
   hom-inv-symmetric-group-loop-group-Set :
     type-hom-Group (symmetric-Group X) (loop-group-Set X)
@@ -129,7 +129,7 @@ module _
   pr2 hom-inv-symmetric-group-loop-group-Set f g =
     ( ap
       ( inv)
-      ( inv (compute-eq-equiv (type-Set X) (type-Set X) (type-Set X) g f))) ∙
+      ( inv (compute-eq-equiv-equiv-comp (type-Set X) (type-Set X) (type-Set X) g f))) ∙
       ( distributive-inv-concat
         ( eq-equiv (type-Set X) (type-Set X) g)
         ( eq-equiv (type-Set X) (type-Set X) f))

From 1843d289642cb7e970c39b38e5557c135b6504bc Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 10:46:13 +0200
Subject: [PATCH 02/37] some variable renamings `ap-eq-equiv`

---
 .../action-on-equivalences-functions.lagda.md      | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index 336fcbe626..955a524be9 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -32,14 +32,14 @@ open import foundation-core.injective-maps
 
 ```agda
 ap-eq-equiv :
-  {l1 l2 : Level} {A : UU l2} (B : UU l1 → A) {X Y : UU l1} →
-  X ≃ Y → B X ＝ B Y
-ap-eq-equiv B {X} {Y} e = ap B (eq-equiv X Y e)
+  {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y : UU l1} →
+  X ≃ Y → f X ＝ f Y
+ap-eq-equiv f {X} {Y} e = ap f (eq-equiv X Y e)
 
 ap-equiv :
-  {l1 l2 : Level} (B : UU l1 → UU l2) {X Y : UU l1} →
-  X ≃ Y → B X ≃ B Y
-ap-equiv B = equiv-eq ∘ ap-eq-equiv B
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
+  X ≃ Y → f X ≃ f Y
+ap-equiv f = equiv-eq ∘ ap-eq-equiv f
 ```
 
 ## Properties
@@ -57,7 +57,7 @@ ap-equiv-id {l} {X} {Y} e =
 
 ```agda
 ap-eq-equiv-comp :
-  {l1 l2 l3 : Level} {A : UU l3} (g : UU l2 → A) (f : UU l1 → UU l2)
+  {l1 l2 l3 : Level} {C : UU l3} (g : UU l2 → C) (f : UU l1 → UU l2)
   {X Y : UU l1} → ap-eq-equiv (g ∘ f) ~ (ap-eq-equiv g ∘ ap-equiv f)
 ap-eq-equiv-comp g f {X} {Y} e =
   ( ap-comp g f (eq-equiv X Y e)) ∙

From f5d62b0d8bbe5ca2cc3f05a35fcb2b661b83b4e6 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 11:32:39 +0200
Subject: [PATCH 03/37] small refactor transport equiv

---
 .../equality-dependent-pair-types.lagda.md     |  4 ++--
 src/foundation/transport.lagda.md              | 18 +++++++++---------
 2 files changed, 11 insertions(+), 11 deletions(-)

diff --git a/src/foundation/equality-dependent-pair-types.lagda.md b/src/foundation/equality-dependent-pair-types.lagda.md
index 6398fdf073..bfab09dcbb 100644
--- a/src/foundation/equality-dependent-pair-types.lagda.md
+++ b/src/foundation/equality-dependent-pair-types.lagda.md
@@ -100,12 +100,12 @@ module _
     {x y : X} (p : x ＝ y) →
     pair-eq-Σ (ap f p) ＝
     ( ( ap (pr1 ∘ f) p) ,
-      ( tr-subst B (pr1 ∘ f) p ∙ apd (λ x → pr2 (f x)) p))
+      ( tr-subst B (pr1 ∘ f) p ∙ apd (pr2 ∘ f) p))
   pair-eq-Σ-ap refl = refl
 
   pr1-pair-eq-Σ-ap :
     {x y : X} (p : x ＝ y) →
-    pr1 (pair-eq-Σ (ap f p)) ＝ ap (λ x → pr1 (f x)) p
+    pr1 (pair-eq-Σ (ap f p)) ＝ ap (pr1 ∘ f) p
   pr1-pair-eq-Σ-ap refl = refl
 ```
 
diff --git a/src/foundation/transport.lagda.md b/src/foundation/transport.lagda.md
index 7cdcfef588..8320807035 100644
--- a/src/foundation/transport.lagda.md
+++ b/src/foundation/transport.lagda.md
@@ -54,21 +54,21 @@ module _
   {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {x y : A}
   where
 
-  inv-tr : x ＝ y → B y → B x
-  inv-tr p = tr B (inv p)
+  tr-inv : x ＝ y → B y → B x
+  tr-inv p = tr B (inv p)
 
-  is-retraction-inv-tr : (p : x ＝ y) → ((inv-tr p) ∘ (tr B p)) ~ id
-  is-retraction-inv-tr refl b = refl
+  is-section-tr : (p : x ＝ y) → (tr-inv p ∘ tr B p) ~ id
+  is-section-tr refl b = refl
 
-  is-section-inv-tr : (p : x ＝ y) → ((tr B p) ∘ (inv-tr p)) ~ id
-  is-section-inv-tr refl b = refl
+  is-retraction-tr : (p : x ＝ y) → (tr B p ∘ tr-inv p) ~ id
+  is-retraction-tr refl b = refl
 
   is-equiv-tr : (p : x ＝ y) → is-equiv (tr B p)
   is-equiv-tr p =
     is-equiv-has-inverse
-      ( inv-tr p)
-      ( is-section-inv-tr p)
-      ( is-retraction-inv-tr p)
+      ( tr-inv p)
+      ( is-retraction-tr p)
+      ( is-section-tr p)
 
   equiv-tr : x ＝ y → (B x) ≃ (B y)
   pr1 (equiv-tr p) = tr B p

From 52af2916c8afc3c87decbb802ea4a3d8e221087d Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 12:05:08 +0200
Subject: [PATCH 04/37] transport along equivalences

---
 src/foundation.lagda.md                       |   1 +
 .../action-on-equivalences-functions.lagda.md |   6 +-
 .../transport-along-equivalences.lagda.md     | 111 ++++++++++++++++++
 src/modal-type-theory.lagda.md                |   0
 4 files changed, 115 insertions(+), 3 deletions(-)
 create mode 100644 src/foundation/transport-along-equivalences.lagda.md
 create mode 100644 src/modal-type-theory.lagda.md

diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index 0e4c08a344..e8133645a5 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -255,6 +255,7 @@ open import foundation.symmetric-operations public
 open import foundation.tight-apartness-relations public
 open import foundation.torsorial-type-families public
 open import foundation.transport public
+open import foundation.transport-along-equivalences public
 open import foundation.trivial-relaxed-sigma-decompositions public
 open import foundation.trivial-sigma-decompositions public
 open import foundation.truncated-equality public
diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index 955a524be9..26623b33f2 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -7,21 +7,21 @@ module foundation.action-on-equivalences-functions where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.univalence
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
 open import foundation.equivalences
 open import foundation.function-extensionality
 open import foundation.fundamental-theorem-of-identity-types
+open import foundation.transport
+open import foundation.univalence
 open import foundation.universe-levels
 
-open import foundation-core.contractible-types
 open import foundation-core.constant-maps
+open import foundation-core.contractible-types
 open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
-open import foundation.transport
 open import foundation-core.identity-types
 open import foundation-core.injective-maps
 ```
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
new file mode 100644
index 0000000000..2147ac1c52
--- /dev/null
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -0,0 +1,111 @@
+# Transport along equivalences
+
+```agda
+module foundation.transport-along-equivalences where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.univalence
+open import foundation.action-on-identifications-functions
+open import foundation.action-on-equivalences-functions
+open import foundation.dependent-pair-types
+open import foundation.equality-dependent-function-types
+open import foundation.equivalences
+open import foundation.fundamental-theorem-of-identity-types
+open import foundation.universe-levels
+
+open import foundation-core.contractible-types
+open import foundation-core.function-types
+open import foundation-core.functoriality-dependent-pair-types
+open import foundation-core.homotopies
+open import foundation.transport
+open import foundation-core.identity-types
+open import foundation-core.injective-maps
+```
+
+</details>
+
+## Definition
+
+```agda
+tr-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
+  X ≃ Y → f X → f Y
+tr-equiv f {X} {Y} e = tr f (eq-equiv X Y e)
+```
+
+## Properties
+
+### Transport along an equivalence is an equivalence
+
+```agda
+is-equiv-tr-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1}
+  (e : X ≃ Y) → is-equiv (tr-equiv f e)
+is-equiv-tr-equiv f {X} {Y} e = is-equiv-tr f (eq-equiv X Y e)
+
+equiv-tr-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
+  X ≃ Y → (f X) ≃ (f Y)
+pr1 (equiv-tr-equiv f e) = tr-equiv f e
+pr2 (equiv-tr-equiv f e) = is-equiv-tr-equiv f e
+```
+
+### Transporting along `id-equiv` is the identity equivalence
+
+```agda
+equiv-tr-equiv-id-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X : UU l1} →
+  equiv-tr-equiv f id-equiv ＝ id-equiv
+equiv-tr-equiv-id-equiv f {X} =
+  ap (equiv-tr f) (compute-eq-equiv-id-equiv X) ∙ equiv-tr-refl f
+```
+
+### Transport along equivalences preserves composition of equivalences
+
+```agda
+tr-equiv-equiv-comp :
+  {l1 l2 : Level} (f : UU l1 → UU l2)
+  {X Y Z : UU l1} (e : X ≃ Y) (e' : Y ≃ Z) →
+  tr-equiv f (e' ∘e e) ~ (tr-equiv f e' ∘ tr-equiv f e)
+tr-equiv-equiv-comp f {X} {Y} {Z} e e' x =
+  ( ap (λ p → tr f p x) (inv (compute-eq-equiv-equiv-comp X Y Z e e'))) ∙
+  ( tr-concat (eq-equiv X Y e) (eq-equiv Y Z e') x)
+```
+
+### Transposing transport along the inverse of an equivalence
+
+```agda
+eq-transpose-tr-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2)
+  {X Y : UU l1} (e : X ≃ Y) {u : f X} {v : f Y} →
+  v ＝ tr-equiv f e u → tr-equiv f (inv-equiv e) v ＝ u
+eq-transpose-tr-equiv f {X} {Y} e {u} {v} p =
+  ( ap (tr-equiv f (inv-equiv e)) p) ∙
+  ( ( ap
+      ( λ q → tr f q (tr-equiv f e u))
+      ( inv (commutativity-inv-eq-equiv X Y e))) ∙
+    ( is-section-tr f (eq-equiv X Y e) u))
+
+eq-transpose-tr-equiv' :
+  {l1 l2 : Level} (f : UU l1 → UU l2)
+  {X Y : UU l1} (e : X ≃ Y) {u : f X} {v : f Y} →
+  tr-equiv f e u ＝ v → u ＝ tr-equiv f (inv-equiv e) v
+eq-transpose-tr-equiv' f {X} {Y} e {u} {v} p =
+  ( inv (is-section-tr f (eq-equiv X Y e) u)) ∙
+  ( ( ap (tr f (inv (eq-equiv X Y e))) p) ∙
+    ( ap (λ q → tr f q v) (commutativity-inv-eq-equiv X Y e)))
+```
+
+### Substitution law for transport along equivalences
+
+```agda
+tr-equiv-subst :
+  {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2) {X Y : UU l1}
+  (e : X ≃ Y) → tr-equiv g (ap-equiv f e) ~ tr-equiv (g ∘ f) e
+tr-equiv-subst g f {X} {Y} e X' =
+  ( ap (λ p → tr g p X') (is-retraction-eq-equiv (ap-eq-equiv f e))) ∙
+  ( tr-subst g f (eq-equiv X Y e))
+```
diff --git a/src/modal-type-theory.lagda.md b/src/modal-type-theory.lagda.md
new file mode 100644
index 0000000000..e69de29bb2

From dc57d3286617005e30f10d7d14348dcc1a4eaddc Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 12:06:06 +0200
Subject: [PATCH 05/37] pre-commit

---
 src/foundation/transport-along-equivalences.lagda.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 2147ac1c52..c646e8dfa2 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -7,20 +7,20 @@ module foundation.transport-along-equivalences where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.univalence
-open import foundation.action-on-identifications-functions
 open import foundation.action-on-equivalences-functions
+open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
 open import foundation.equivalences
 open import foundation.fundamental-theorem-of-identity-types
+open import foundation.transport
+open import foundation.univalence
 open import foundation.universe-levels
 
 open import foundation-core.contractible-types
 open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
-open import foundation.transport
 open import foundation-core.identity-types
 open import foundation-core.injective-maps
 ```

From c3f1a4d3d4d7882a8467b10e709a8de772d330ee Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 14:10:09 +0200
Subject: [PATCH 06/37] character limit

---
 src/group-theory/loop-groups-sets.lagda.md | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/src/group-theory/loop-groups-sets.lagda.md b/src/group-theory/loop-groups-sets.lagda.md
index cb6109b443..5aa1df7846 100644
--- a/src/group-theory/loop-groups-sets.lagda.md
+++ b/src/group-theory/loop-groups-sets.lagda.md
@@ -129,7 +129,9 @@ module _
   pr2 hom-inv-symmetric-group-loop-group-Set f g =
     ( ap
       ( inv)
-      ( inv (compute-eq-equiv-equiv-comp (type-Set X) (type-Set X) (type-Set X) g f))) ∙
+      ( inv
+        ( compute-eq-equiv-equiv-comp
+          ( type-Set X) (type-Set X) (type-Set X) g f))) ∙
       ( distributive-inv-concat
         ( eq-equiv (type-Set X) (type-Set X) g)
         ( eq-equiv (type-Set X) (type-Set X) f))

From 35c3f966cced7580e4140fd3f9f219d8fd073df3 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 14:29:48 +0200
Subject: [PATCH 07/37] missed renamings

---
 .../universal-property-set-quotients.lagda.md        | 12 ++++++------
 src/lists/permutation-lists.lagda.md                 |  2 +-
 2 files changed, 7 insertions(+), 7 deletions(-)

diff --git a/src/foundation/universal-property-set-quotients.lagda.md b/src/foundation/universal-property-set-quotients.lagda.md
index 91a6a1ad24..203fa60492 100644
--- a/src/foundation/universal-property-set-quotients.lagda.md
+++ b/src/foundation/universal-property-set-quotients.lagda.md
@@ -426,11 +426,11 @@ module _
     α : Σ (A → Prop l2) (reflects-Equivalence-Relation R)
     α = pair
         ( prop-Equivalence-Relation R x)
-          ( λ r →
-            eq-iff
-              ( transitive-Equivalence-Relation R _ _ _ r)
-              ( transitive-Equivalence-Relation R _ _ _
-                ( symmetric-Equivalence-Relation R _ _ r)))
+        ( λ r →
+          eq-iff
+            ( transitive-Equivalence-Relation R _ _ _ r)
+            ( transitive-Equivalence-Relation R _ _ _
+              ( symmetric-Equivalence-Relation R _ _ r)))
     P : type-Set B → Prop l2
     P = map-inv-is-equiv (Q (Prop-Set l2)) α
     compute-P :
@@ -460,7 +460,7 @@ module _
             ( reflects-map-reflecting-map-Equivalence-Relation R q
               ( map-inv-equiv
                 ( compute-P (pr1 v))
-                ( inv-tr (λ b → type-Prop (P b)) (pr2 v) p))) ∙
+                ( tr-inv (type-Prop ∘ P) (pr2 v) p))) ∙
             ( pr2 v)))
     is-contr-total-P : is-contr (Σ (type-Set B) (λ b → type-Prop (P b)))
     is-contr-total-P = pair center-total-P contraction-total-P
diff --git a/src/lists/permutation-lists.lagda.md b/src/lists/permutation-lists.lagda.md
index 32fb571f8d..cab237461e 100644
--- a/src/lists/permutation-lists.lagda.md
+++ b/src/lists/permutation-lists.lagda.md
@@ -289,7 +289,7 @@ eq-map-list-permute-list {B = B} f p t =
               ( t)) ∙
             ( ap
               ( λ v → permute-vec (length-list p) v t)
-              ( is-section-inv-tr
+              ( is-retraction-tr
                 ( vec B)
                 ( length-map-list f p)
                 ( map-vec f (vec-list p)))))))) ∙

From 25d2ff3148f68a3f4fe0a0c7cd57ce51ae645419 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 14:35:23 +0200
Subject: [PATCH 08/37] `transport` -> `transport-along-identifications`

---
 .../bezouts-lemma-integers.lagda.md                   |  2 +-
 .../bezouts-lemma-natural-numbers.lagda.md            |  2 +-
 src/elementary-number-theory/cofibonacci.lagda.md     |  2 +-
 .../congruence-integers.lagda.md                      |  2 +-
 .../congruence-natural-numbers.lagda.md               |  2 +-
 .../distance-natural-numbers.lagda.md                 |  2 +-
 .../divisibility-integers.lagda.md                    |  2 +-
 .../divisibility-natural-numbers.lagda.md             |  2 +-
 .../fibonacci-sequence.lagda.md                       |  2 +-
 .../fundamental-theorem-of-arithmetic.lagda.md        |  2 +-
 .../greatest-common-divisor-integers.lagda.md         |  2 +-
 .../greatest-common-divisor-natural-numbers.lagda.md  |  2 +-
 .../inequality-integers.lagda.md                      |  2 +-
 src/elementary-number-theory/powers-of-two.lagda.md   |  2 +-
 src/elementary-number-theory/prime-numbers.lagda.md   |  2 +-
 .../reduced-integer-fractions.lagda.md                |  2 +-
 .../relatively-prime-natural-numbers.lagda.md         |  2 +-
 .../strict-inequality-natural-numbers.lagda.md        |  2 +-
 ...-ordering-principle-standard-finite-types.lagda.md |  2 +-
 .../cartier-delooping-sign-homomorphism.lagda.md      |  2 +-
 .../concrete-quaternion-group.lagda.md                |  2 +-
 .../delooping-sign-homomorphism.lagda.md              |  2 +-
 src/finite-group-theory/orbits-permutations.lagda.md  |  2 +-
 .../simpson-delooping-sign-homomorphism.lagda.md      |  2 +-
 src/finite-group-theory/transpositions.lagda.md       |  2 +-
 src/foundation-core.lagda.md                          |  2 +-
 src/foundation-core/contractible-types.lagda.md       |  2 +-
 .../dependent-identifications.lagda.md                | 11 ++++++-----
 .../equality-dependent-pair-types.lagda.md            |  2 +-
 src/foundation-core/fibers-of-maps.lagda.md           |  2 +-
 .../functoriality-dependent-function-types.lagda.md   |  2 +-
 .../functoriality-dependent-pair-types.lagda.md       |  2 +-
 src/foundation-core/homotopies.lagda.md               |  2 +-
 src/foundation-core/identity-types.lagda.md           |  4 ++--
 src/foundation-core/propositions.lagda.md             |  2 +-
 src/foundation-core/singleton-induction.lagda.md      |  2 +-
 src/foundation-core/small-types.lagda.md              |  2 +-
 src/foundation-core/subtypes.lagda.md                 |  2 +-
 ...da.md => transport-along-identifications.lagda.md} |  7 ++++---
 src/foundation-core/truncated-types.lagda.md          |  2 +-
 src/foundation-core/univalence.lagda.md               |  2 +-
 src/foundation.lagda.md                               |  2 +-
 .../action-on-equivalences-functions.lagda.md         |  2 +-
 ...on-on-identifications-dependent-functions.lagda.md |  6 +++---
 .../action-on-identifications-functions.lagda.md      |  2 +-
 ...ic-law-coproduct-and-sigma-decompositions.lagda.md |  2 +-
 ...thmetic-law-product-and-pi-decompositions.lagda.md |  2 +-
 src/foundation/binary-transport.lagda.md              |  2 +-
 src/foundation/cartesian-product-types.lagda.md       |  2 +-
 src/foundation/cones-over-cospans.lagda.md            |  2 +-
 src/foundation/coproduct-decompositions.lagda.md      |  2 +-
 src/foundation/decidable-equality.lagda.md            |  2 +-
 .../decidable-equivalence-relations.lagda.md          |  2 +-
 src/foundation/decidable-propositions.lagda.md        |  2 +-
 src/foundation/decidable-subtypes.lagda.md            |  2 +-
 src/foundation/dependent-binomial-theorem.lagda.md    |  2 +-
 src/foundation/dependent-identifications.lagda.md     |  2 +-
 src/foundation/double-negation-modality.lagda.md      |  2 +-
 .../equality-cartesian-product-types.lagda.md         |  2 +-
 src/foundation/equality-dependent-pair-types.lagda.md |  2 +-
 src/foundation/equality-fibers-of-maps.lagda.md       |  2 +-
 src/foundation/equivalence-relations.lagda.md         |  2 +-
 src/foundation/exclusive-disjunction.lagda.md         |  2 +-
 src/foundation/fiber-inclusions.lagda.md              |  2 +-
 src/foundation/full-subtypes.lagda.md                 |  2 +-
 src/foundation/function-types.lagda.md                |  2 +-
 src/foundation/functoriality-coproduct-types.lagda.md |  2 +-
 .../functoriality-dependent-function-types.lagda.md   |  2 +-
 .../functoriality-dependent-pair-types.lagda.md       |  2 +-
 src/foundation/homotopies.lagda.md                    |  2 +-
 src/foundation/identity-systems.lagda.md              |  2 +-
 src/foundation/identity-types.lagda.md                |  6 +++---
 ...uction-principle-propositional-truncation.lagda.md |  2 +-
 src/foundation/isolated-points.lagda.md               |  2 +-
 src/foundation/locally-small-types.lagda.md           |  2 +-
 src/foundation/partitions.lagda.md                    |  2 +-
 src/foundation/perfect-images.lagda.md                |  2 +-
 .../pointed-torsorial-type-families.lagda.md          |  2 +-
 .../products-unordered-pairs-of-types.lagda.md        |  2 +-
 src/foundation/propositional-extensionality.lagda.md  |  2 +-
 src/foundation/propositional-truncations.lagda.md     |  2 +-
 src/foundation/pullbacks.lagda.md                     |  2 +-
 src/foundation/singleton-induction.lagda.md           |  2 +-
 src/foundation/structure.lagda.md                     |  2 +-
 src/foundation/subuniverses.lagda.md                  |  2 +-
 src/foundation/symmetric-difference.lagda.md          |  2 +-
 src/foundation/transport-along-equivalences.lagda.md  |  2 +-
 ...da.md => transport-along-identifications.lagda.md} | 11 ++++++-----
 .../type-theoretic-principle-of-choice.lagda.md       |  2 +-
 .../univalence-action-on-equivalences.lagda.md        |  2 +-
 ...nivalence-implies-function-extensionality.lagda.md |  2 +-
 src/foundation/univalent-type-families.lagda.md       |  2 +-
 src/foundation/universal-property-image.lagda.md      |  2 +-
 .../universal-property-set-quotients.lagda.md         |  2 +-
 .../equivalences-enriched-undirected-graphs.lagda.md  |  2 +-
 .../equivalences-undirected-graphs.lagda.md           |  2 +-
 src/graph-theory/fibers-directed-graphs.lagda.md      |  2 +-
 src/graph-theory/morphisms-undirected-graphs.lagda.md |  2 +-
 src/graph-theory/undirected-graphs.lagda.md           |  2 +-
 src/graph-theory/walks-directed-graphs.lagda.md       |  2 +-
 src/group-theory/concrete-group-actions.lagda.md      |  2 +-
 .../homomorphisms-concrete-group-actions.lagda.md     |  2 +-
 src/group-theory/normal-subgroups.lagda.md            |  2 +-
 src/group-theory/subgroups.lagda.md                   |  2 +-
 src/group-theory/torsors.lagda.md                     |  2 +-
 .../transitive-concrete-group-actions.lagda.md        |  2 +-
 src/higher-group-theory/higher-group-actions.lagda.md |  2 +-
 .../homomorphisms-higher-group-actions.lagda.md       |  2 +-
 src/linear-algebra/vectors.lagda.md                   |  2 +-
 src/lists/functoriality-lists.lagda.md                |  2 +-
 src/lists/permutation-lists.lagda.md                  |  2 +-
 src/lists/permutation-vectors.lagda.md                |  2 +-
 src/lists/sort-by-insertion-vectors.lagda.md          |  2 +-
 src/order-theory/finitely-graded-posets.lagda.md      |  2 +-
 src/organic-chemistry/ethane.lagda.md                 |  2 +-
 .../extensions-of-maps.lagda.md                       |  2 +-
 .../open-modalities.lagda.md                          |  2 +-
 src/polytopes/abstract-polytopes.lagda.md             |  2 +-
 .../ideals-generated-by-subsets-rings.lagda.md        |  2 +-
 .../left-ideals-generated-by-subsets-rings.lagda.md   |  2 +-
 .../right-ideals-generated-by-subsets-rings.lagda.md  |  2 +-
 src/set-theory/cumulative-hierarchy.lagda.md          |  2 +-
 ...position-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...nentials-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...products-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...nentials-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...products-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...position-species-of-types-in-subuniverses.lagda.md |  2 +-
 src/species/species-of-types-in-subuniverses.lagda.md |  2 +-
 src/species/species-of-types.lagda.md                 |  2 +-
 src/structured-types/pointed-maps.lagda.md            |  2 +-
 src/synthetic-homotopy-theory/26-descent.lagda.md     |  2 +-
 src/synthetic-homotopy-theory/26-id-pushout.lagda.md  |  2 +-
 src/synthetic-homotopy-theory/circle.lagda.md         |  2 +-
 .../dependent-cocones-under-spans.lagda.md            |  2 +-
 .../dependent-pullback-property-pushouts.lagda.md     |  2 +-
 .../dependent-universal-property-pushouts.lagda.md    |  2 +-
 src/synthetic-homotopy-theory/descent-circle.lagda.md |  2 +-
 src/synthetic-homotopy-theory/free-loops.lagda.md     |  2 +-
 src/synthetic-homotopy-theory/interval-type.lagda.md  |  2 +-
 .../multiplication-circle.lagda.md                    |  2 +-
 .../suspensions-of-types.lagda.md                     |  2 +-
 .../universal-cover-circle.lagda.md                   |  2 +-
 .../universal-property-circle.lagda.md                |  2 +-
 src/trees/bases-directed-trees.lagda.md               |  2 +-
 src/trees/directed-trees.lagda.md                     |  2 +-
 src/trees/elementhood-relation-w-types.lagda.md       |  2 +-
 src/trees/enriched-directed-trees.lagda.md            |  2 +-
 src/trees/equivalences-directed-trees.lagda.md        |  2 +-
 .../equivalences-enriched-directed-trees.lagda.md     |  2 +-
 src/trees/extensional-w-types.lagda.md                |  2 +-
 .../functoriality-combinator-directed-trees.lagda.md  |  2 +-
 src/trees/functoriality-w-types.lagda.md              |  2 +-
 src/trees/induction-w-types.lagda.md                  |  2 +-
 src/trees/morphisms-directed-trees.lagda.md           |  2 +-
 src/trees/morphisms-enriched-directed-trees.lagda.md  |  2 +-
 src/trees/polynomial-endofunctors.lagda.md            |  2 +-
 src/trees/ranks-of-elements-w-types.lagda.md          |  2 +-
 src/trees/rooted-morphisms-directed-trees.lagda.md    |  2 +-
 .../rooted-morphisms-enriched-directed-trees.lagda.md |  2 +-
 src/trees/small-multisets.lagda.md                    |  2 +-
 ...ements-coalgebras-polynomial-endofunctors.lagda.md |  2 +-
 .../underlying-trees-of-elements-of-w-types.lagda.md  |  2 +-
 src/trees/undirected-trees.lagda.md                   |  2 +-
 src/trees/universal-multiset.lagda.md                 |  2 +-
 src/trees/w-types.lagda.md                            |  2 +-
 src/type-theories/dependent-type-theories.lagda.md    |  2 +-
 .../fibered-dependent-type-theories.lagda.md          |  2 +-
 .../sections-dependent-type-theories.lagda.md         |  2 +-
 src/type-theories/simple-type-theories.lagda.md       |  2 +-
 .../2-element-decidable-subtypes.lagda.md             |  2 +-
 .../2-element-subtypes.lagda.md                       |  2 +-
 src/univalent-combinatorics/2-element-types.lagda.md  |  2 +-
 .../counting-dependent-pair-types.lagda.md            |  2 +-
 ...cycle-prime-decomposition-natural-numbers.lagda.md |  2 +-
 .../decidable-subtypes.lagda.md                       |  2 +-
 .../dependent-pair-types.lagda.md                     |  2 +-
 .../equivalences-cubes.lagda.md                       |  2 +-
 src/univalent-combinatorics/fibers-of-maps.lagda.md   |  2 +-
 src/univalent-combinatorics/finite-types.lagda.md     |  2 +-
 .../orientations-complete-undirected-graph.lagda.md   |  2 +-
 src/univalent-combinatorics/pi-finite-types.lagda.md  |  2 +-
 src/universal-algebra/terms-over-signatures.lagda.md  |  2 +-
 183 files changed, 201 insertions(+), 198 deletions(-)
 rename src/foundation-core/{transport.lagda.md => transport-along-identifications.lagda.md} (89%)
 rename src/foundation/{transport.lagda.md => transport-along-identifications.lagda.md} (85%)

diff --git a/src/elementary-number-theory/bezouts-lemma-integers.lagda.md b/src/elementary-number-theory/bezouts-lemma-integers.lagda.md
index 536b969aa3..b94ea9b729 100644
--- a/src/elementary-number-theory/bezouts-lemma-integers.lagda.md
+++ b/src/elementary-number-theory/bezouts-lemma-integers.lagda.md
@@ -28,7 +28,7 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 ```
 
diff --git a/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md b/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md
index 19ad5b7dc7..ad127678d2 100755
--- a/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md
@@ -37,7 +37,7 @@ open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import univalent-combinatorics.standard-finite-types
diff --git a/src/elementary-number-theory/cofibonacci.lagda.md b/src/elementary-number-theory/cofibonacci.lagda.md
index e80bd5387e..8d1f91db9b 100644
--- a/src/elementary-number-theory/cofibonacci.lagda.md
+++ b/src/elementary-number-theory/cofibonacci.lagda.md
@@ -22,7 +22,7 @@ open import foundation.decidable-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/congruence-integers.lagda.md b/src/elementary-number-theory/congruence-integers.lagda.md
index 51d82b086d..9844348d01 100644
--- a/src/elementary-number-theory/congruence-integers.lagda.md
+++ b/src/elementary-number-theory/congruence-integers.lagda.md
@@ -22,7 +22,7 @@ open import foundation.binary-relations
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/congruence-natural-numbers.lagda.md b/src/elementary-number-theory/congruence-natural-numbers.lagda.md
index 732c75a31d..57781d53a3 100644
--- a/src/elementary-number-theory/congruence-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/congruence-natural-numbers.lagda.md
@@ -19,7 +19,7 @@ open import foundation.binary-relations
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import univalent-combinatorics.standard-finite-types
diff --git a/src/elementary-number-theory/distance-natural-numbers.lagda.md b/src/elementary-number-theory/distance-natural-numbers.lagda.md
index 4aa61b2b9e..0426593e35 100644
--- a/src/elementary-number-theory/distance-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/distance-natural-numbers.lagda.md
@@ -19,7 +19,7 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/divisibility-integers.lagda.md b/src/elementary-number-theory/divisibility-integers.lagda.md
index e570a1a875..b6c1fc7453 100644
--- a/src/elementary-number-theory/divisibility-integers.lagda.md
+++ b/src/elementary-number-theory/divisibility-integers.lagda.md
@@ -27,7 +27,7 @@ open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositional-maps
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md
index 97ce7e55af..869a1a4189 100644
--- a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md
@@ -23,7 +23,7 @@ open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-maps
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/elementary-number-theory/fibonacci-sequence.lagda.md b/src/elementary-number-theory/fibonacci-sequence.lagda.md
index ed4d4e50cf..634cc34706 100644
--- a/src/elementary-number-theory/fibonacci-sequence.lagda.md
+++ b/src/elementary-number-theory/fibonacci-sequence.lagda.md
@@ -17,7 +17,7 @@ open import elementary-number-theory.relatively-prime-natural-numbers
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 ```
 
 </details>
diff --git a/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md b/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md
index b05cff6d00..992aa4a2fa 100644
--- a/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md
+++ b/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md
@@ -35,7 +35,7 @@ open import foundation.empty-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md b/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md
index 8fa61c143e..2dc2e9236a 100644
--- a/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md
+++ b/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md
@@ -24,7 +24,7 @@ open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md b/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md
index 32b773f17b..8b4fdaaa35 100644
--- a/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md
@@ -30,7 +30,7 @@ open import foundation.empty-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/inequality-integers.lagda.md b/src/elementary-number-theory/inequality-integers.lagda.md
index 2955f6bf44..aa92163af9 100644
--- a/src/elementary-number-theory/inequality-integers.lagda.md
+++ b/src/elementary-number-theory/inequality-integers.lagda.md
@@ -19,7 +19,7 @@ open import foundation.function-types
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/powers-of-two.lagda.md b/src/elementary-number-theory/powers-of-two.lagda.md
index d1b2c00db1..9efb3ca6ba 100644
--- a/src/elementary-number-theory/powers-of-two.lagda.md
+++ b/src/elementary-number-theory/powers-of-two.lagda.md
@@ -21,7 +21,7 @@ open import foundation.dependent-pair-types
 open import foundation.equality-cartesian-product-types
 open import foundation.function-types
 open import foundation.split-surjective-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.cartesian-product-types
diff --git a/src/elementary-number-theory/prime-numbers.lagda.md b/src/elementary-number-theory/prime-numbers.lagda.md
index ad4ce088ba..00cb8b2065 100644
--- a/src/elementary-number-theory/prime-numbers.lagda.md
+++ b/src/elementary-number-theory/prime-numbers.lagda.md
@@ -28,7 +28,7 @@ open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/reduced-integer-fractions.lagda.md b/src/elementary-number-theory/reduced-integer-fractions.lagda.md
index a6dd71808f..06aa1843e5 100644
--- a/src/elementary-number-theory/reduced-integer-fractions.lagda.md
+++ b/src/elementary-number-theory/reduced-integer-fractions.lagda.md
@@ -26,7 +26,7 @@ open import foundation.equality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md b/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md
index 61c91bb897..fbf54cb39d 100644
--- a/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md
@@ -20,7 +20,7 @@ open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md
index 847e67d95c..a70e22d17a 100644
--- a/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md
@@ -22,7 +22,7 @@ open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md b/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md
index e392a72dba..3e7b58ad9d 100644
--- a/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md
+++ b/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md
@@ -31,7 +31,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-empty-type
 open import foundation.type-arithmetic-unit-type
diff --git a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
index 0271529c3a..18ce157974 100644
--- a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
@@ -27,7 +27,7 @@ open import foundation.mere-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.univalence-action-on-equivalences
 open import foundation.universe-levels
diff --git a/src/finite-group-theory/concrete-quaternion-group.lagda.md b/src/finite-group-theory/concrete-quaternion-group.lagda.md
index 0c66d78790..d03507a4e9 100644
--- a/src/finite-group-theory/concrete-quaternion-group.lagda.md
+++ b/src/finite-group-theory/concrete-quaternion-group.lagda.md
@@ -13,7 +13,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.isolated-points
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import univalent-combinatorics.complements-isolated-points
diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index 951248c1cb..8034a561a9 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -47,7 +47,7 @@ open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.reflecting-maps-equivalence-relations
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.uniqueness-set-quotients
 open import foundation.unit-type
diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md
index 0d3195117e..5e5fef1f25 100644
--- a/src/finite-group-theory/orbits-permutations.lagda.md
+++ b/src/finite-group-theory/orbits-permutations.lagda.md
@@ -48,7 +48,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.repetitions-of-values
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
index 2efe2fbeb5..1166b556df 100644
--- a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
@@ -41,7 +41,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.univalence-action-on-equivalences
 open import foundation.universe-levels
diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md
index 02d1b40715..06b4c99881 100644
--- a/src/finite-group-theory/transpositions.lagda.md
+++ b/src/finite-group-theory/transpositions.lagda.md
@@ -40,7 +40,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/foundation-core.lagda.md b/src/foundation-core.lagda.md
index 72aaa8803c..8298257aca 100644
--- a/src/foundation-core.lagda.md
+++ b/src/foundation-core.lagda.md
@@ -48,7 +48,7 @@ open import foundation-core.sets public
 open import foundation-core.singleton-induction public
 open import foundation-core.small-types public
 open import foundation-core.subtypes public
-open import foundation-core.transport public
+open import foundation-core.transport-along-identifications public
 open import foundation-core.truncated-maps public
 open import foundation-core.truncated-types public
 open import foundation-core.truncation-levels public
diff --git a/src/foundation-core/contractible-types.lagda.md b/src/foundation-core/contractible-types.lagda.md
index cfa56145e5..73fc88a726 100644
--- a/src/foundation-core/contractible-types.lagda.md
+++ b/src/foundation-core/contractible-types.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.equality-dependent-pair-types
 open import foundation-core.equivalences
 open import foundation-core.identity-types
 open import foundation-core.retractions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/dependent-identifications.lagda.md b/src/foundation-core/dependent-identifications.lagda.md
index 9a5da82082..55178798aa 100644
--- a/src/foundation-core/dependent-identifications.lagda.md
+++ b/src/foundation-core/dependent-identifications.lagda.md
@@ -10,7 +10,7 @@ module foundation-core.dependent-identifications where
 open import foundation.universe-levels
 
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
@@ -26,10 +26,11 @@ identification
   tr B p u ＝ v,
 ```
 
-where `tr` is the [transport](foundation-core.transport.md) function. Dependent
-identifications also satisfy groupoid laws, which are formulated appropriately
-as dependent identifications. The groupoid laws for dependent identifications
-are proven in
+where `tr` is the
+[transport](foundation-core.transport-along-identifications.md) function.
+Dependent identifications also satisfy groupoid laws, which are formulated
+appropriately as dependent identifications. The groupoid laws for dependent
+identifications are proven in
 [`foundation.dependent-identifications`](foundation.dependent-identifications.md).
 
 ## Definition
diff --git a/src/foundation-core/equality-dependent-pair-types.lagda.md b/src/foundation-core/equality-dependent-pair-types.lagda.md
index d9b33e2faa..f07acb266f 100644
--- a/src/foundation-core/equality-dependent-pair-types.lagda.md
+++ b/src/foundation-core/equality-dependent-pair-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/fibers-of-maps.lagda.md b/src/foundation-core/fibers-of-maps.lagda.md
index 8cffa5184c..5d5c22dbf4 100644
--- a/src/foundation-core/fibers-of-maps.lagda.md
+++ b/src/foundation-core/fibers-of-maps.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/functoriality-dependent-function-types.lagda.md b/src/foundation-core/functoriality-dependent-function-types.lagda.md
index 33ff55f232..a48e86bd2e 100644
--- a/src/foundation-core/functoriality-dependent-function-types.lagda.md
+++ b/src/foundation-core/functoriality-dependent-function-types.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.path-split-maps
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/functoriality-dependent-pair-types.lagda.md b/src/foundation-core/functoriality-dependent-pair-types.lagda.md
index ada200ac6c..8bf0dca52a 100644
--- a/src/foundation-core/functoriality-dependent-pair-types.lagda.md
+++ b/src/foundation-core/functoriality-dependent-pair-types.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.fibers-of-maps
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/homotopies.lagda.md b/src/foundation-core/homotopies.lagda.md
index d94bb57a82..f262a3684d 100644
--- a/src/foundation-core/homotopies.lagda.md
+++ b/src/foundation-core/homotopies.lagda.md
@@ -15,7 +15,7 @@ open import foundation.universe-levels
 open import foundation-core.dependent-identifications
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/identity-types.lagda.md b/src/foundation-core/identity-types.lagda.md
index dd31fadcbf..be603bd50c 100644
--- a/src/foundation-core/identity-types.lagda.md
+++ b/src/foundation-core/identity-types.lagda.md
@@ -67,8 +67,8 @@ identifications in arbitrary types.
 | Path algebra                                     | [`foundation.path-algebra`](foundation.path-algebra.md)                                                                   |
 | Symmetric identity types                         | [`foundation.symmetric-identity-types`](foundation.symmetric-identity-types.md)                                           |
 | Torsorial type families                          | [`foundation.torsorial-type-families`](foundation.torsorial-type-families.md)                                             |
-| Transport (foundation)                           | [`foundation.transport`](foundation.transport.md)                                                                         |
-| Transport (foundation-core)                      | [`foundation-core.transport`](foundation-core.transport.md)                                                               |
+| Transport (foundation)                           | [`foundation.transport-along-identifications`](foundation.transport-along-identifications.md)                             |
+| Transport (foundation-core)                      | [`foundation-core.transport-along-identifications`](foundation-core.transport-along-identifications.md)                   |
 | The universal property of identity types         | [`foundation.universal-property-identity-types`](foundation.universal-property-identity-types.md)                         |
 
 ## Definition
diff --git a/src/foundation-core/propositions.lagda.md b/src/foundation-core/propositions.lagda.md
index 45a5b25af2..9901fae32b 100644
--- a/src/foundation-core/propositions.lagda.md
+++ b/src/foundation-core/propositions.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.function-extensionality
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/singleton-induction.lagda.md b/src/foundation-core/singleton-induction.lagda.md
index aa772e61a2..636f391214 100644
--- a/src/foundation-core/singleton-induction.lagda.md
+++ b/src/foundation-core/singleton-induction.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.sections
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/small-types.lagda.md b/src/foundation-core/small-types.lagda.md
index 0c01b6c305..6f676a6604 100644
--- a/src/foundation-core/small-types.lagda.md
+++ b/src/foundation-core/small-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation.identity-types
 open import foundation.mere-equivalences
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/foundation-core/subtypes.lagda.md b/src/foundation-core/subtypes.lagda.md
index f738fedc79..523a820842 100644
--- a/src/foundation-core/subtypes.lagda.md
+++ b/src/foundation-core/subtypes.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.logical-equivalences
 open import foundation-core.propositional-maps
 open import foundation-core.propositions
 open import foundation-core.sets
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation-core/transport.lagda.md b/src/foundation-core/transport-along-identifications.lagda.md
similarity index 89%
rename from src/foundation-core/transport.lagda.md
rename to src/foundation-core/transport-along-identifications.lagda.md
index a39e2f1e21..176a78d957 100644
--- a/src/foundation-core/transport.lagda.md
+++ b/src/foundation-core/transport-along-identifications.lagda.md
@@ -1,7 +1,7 @@
-# Transport
+# Transport along identifications
 
 ```agda
-module foundation-core.transport where
+module foundation-core.transport-along-identifications where
 ```
 
 <details><summary>Imports</summary>
@@ -22,7 +22,8 @@ element `b : B x`, we can **transport** the element `b` along the identification
 `p` to obtain an element `tr B p b : B y`.
 
 The fact that `tr B p` is an [equivalence](foundation-core.equivalences.md) is
-recorded in [`foundation.transport`](foundation.transport.md).
+recorded in
+[`foundation.transport-along-identifications`](foundation.transport-along-identifications.md).
 
 ## Definition
 
diff --git a/src/foundation-core/truncated-types.lagda.md b/src/foundation-core/truncated-types.lagda.md
index 77b40c7c75..921e8c54f6 100644
--- a/src/foundation-core/truncated-types.lagda.md
+++ b/src/foundation-core/truncated-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.retractions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 open import foundation-core.truncation-levels
 ```
 
diff --git a/src/foundation-core/univalence.lagda.md b/src/foundation-core/univalence.lagda.md
index d906487d4c..3ecbc79a9f 100644
--- a/src/foundation-core/univalence.lagda.md
+++ b/src/foundation-core/univalence.lagda.md
@@ -15,7 +15,7 @@ open import foundation.universe-levels
 open import foundation-core.contractible-types
 open import foundation-core.equivalences
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index e8133645a5..60e18c55fc 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -254,8 +254,8 @@ open import foundation.symmetric-identity-types public
 open import foundation.symmetric-operations public
 open import foundation.tight-apartness-relations public
 open import foundation.torsorial-type-families public
-open import foundation.transport public
 open import foundation.transport-along-equivalences public
+open import foundation.transport-along-identifications public
 open import foundation.trivial-relaxed-sigma-decompositions public
 open import foundation.trivial-sigma-decompositions public
 open import foundation.truncated-equality public
diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index 26623b33f2..75740127a2 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -13,7 +13,7 @@ open import foundation.equality-dependent-function-types
 open import foundation.equivalences
 open import foundation.function-extensionality
 open import foundation.fundamental-theorem-of-identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/foundation/action-on-identifications-dependent-functions.lagda.md b/src/foundation/action-on-identifications-dependent-functions.lagda.md
index e639d4d709..6db5aec11d 100644
--- a/src/foundation/action-on-identifications-dependent-functions.lagda.md
+++ b/src/foundation/action-on-identifications-dependent-functions.lagda.md
@@ -21,9 +21,9 @@ Given a dependent function `f : (x : A) → B x` and an
 [identification](foundation-core.identity-types.md) `p : x ＝ y` in `A`, we
 cannot directly compare the values `f x` and `f y`, since `f x` is an element of
 type `B x` while `f y` is an element of type `B y`. However, we can
-[transport](foundation-core.transport.md) the value `f x` along `p` to obtain an
-element of type `B y` that is comparable to the value `f y`. In other words, we
-can consider the type of
+[transport](foundation-core.transport-along-identifications.md) the value `f x`
+along `p` to obtain an element of type `B y` that is comparable to the value
+`f y`. In other words, we can consider the type of
 [dependent identifications](foundation-core.dependent-identifications.md) over
 `p` from `f x` to `f y`. The **dependent action on identifications** of `f` on
 `p` is a dependent identification
diff --git a/src/foundation/action-on-identifications-functions.lagda.md b/src/foundation/action-on-identifications-functions.lagda.md
index 7ff1b0867d..4fd28c92ed 100644
--- a/src/foundation/action-on-identifications-functions.lagda.md
+++ b/src/foundation/action-on-identifications-functions.lagda.md
@@ -12,7 +12,7 @@ open import foundation.universe-levels
 open import foundation-core.constant-maps
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md b/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md
index 8e76432c07..225b0ebaa9 100644
--- a/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md
+++ b/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md b/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md
index 5a34d34c97..0d028fa529 100644
--- a/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md
+++ b/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/binary-transport.lagda.md b/src/foundation/binary-transport.lagda.md
index 5de9ca3d4f..5ac3a40225 100644
--- a/src/foundation/binary-transport.lagda.md
+++ b/src/foundation/binary-transport.lagda.md
@@ -14,7 +14,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/cartesian-product-types.lagda.md b/src/foundation/cartesian-product-types.lagda.md
index eefc3e3266..1c324495a7 100644
--- a/src/foundation/cartesian-product-types.lagda.md
+++ b/src/foundation/cartesian-product-types.lagda.md
@@ -14,7 +14,7 @@ open import foundation.subuniverses
 open import foundation.universe-levels
 
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/cones-over-cospans.lagda.md b/src/foundation/cones-over-cospans.lagda.md
index 40e0417706..dcb1e94c20 100644
--- a/src/foundation/cones-over-cospans.lagda.md
+++ b/src/foundation/cones-over-cospans.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.contractible-types
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/coproduct-decompositions.lagda.md b/src/foundation/coproduct-decompositions.lagda.md
index f3219be2d7..3bdb3d1749 100644
--- a/src/foundation/coproduct-decompositions.lagda.md
+++ b/src/foundation/coproduct-decompositions.lagda.md
@@ -32,7 +32,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 
 open import univalent-combinatorics.equality-standard-finite-types
 open import univalent-combinatorics.standard-finite-types
diff --git a/src/foundation/decidable-equality.lagda.md b/src/foundation/decidable-equality.lagda.md
index 7e7f86ce21..fd89e09b86 100644
--- a/src/foundation/decidable-equality.lagda.md
+++ b/src/foundation/decidable-equality.lagda.md
@@ -28,7 +28,7 @@ open import foundation-core.injective-maps
 open import foundation-core.propositions
 open import foundation-core.retractions
 open import foundation-core.sets
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/decidable-equivalence-relations.lagda.md b/src/foundation/decidable-equivalence-relations.lagda.md
index 6a2bd8bc09..e61f0003d3 100644
--- a/src/foundation/decidable-equivalence-relations.lagda.md
+++ b/src/foundation/decidable-equivalence-relations.lagda.md
@@ -45,7 +45,7 @@ open import foundation-core.identity-types
 open import foundation-core.logical-equivalences
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/decidable-propositions.lagda.md b/src/foundation/decidable-propositions.lagda.md
index 6bed5895cd..baae1d24a5 100644
--- a/src/foundation/decidable-propositions.lagda.md
+++ b/src/foundation/decidable-propositions.lagda.md
@@ -34,7 +34,7 @@ open import foundation-core.propositions
 open import foundation-core.sets
 open import foundation-core.small-types
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 
 open import univalent-combinatorics.counting
 open import univalent-combinatorics.finite-types
diff --git a/src/foundation/decidable-subtypes.lagda.md b/src/foundation/decidable-subtypes.lagda.md
index 611ae47d43..e4700eb666 100644
--- a/src/foundation/decidable-subtypes.lagda.md
+++ b/src/foundation/decidable-subtypes.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.identity-types
 open import foundation-core.injective-maps
 open import foundation-core.logical-equivalences
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation/dependent-binomial-theorem.lagda.md b/src/foundation/dependent-binomial-theorem.lagda.md
index ccd5203beb..f6e388bcb0 100644
--- a/src/foundation/dependent-binomial-theorem.lagda.md
+++ b/src/foundation/dependent-binomial-theorem.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-dependent-function-types
 open import foundation.raising-universe-levels
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.universal-property-coproduct-types
 open import foundation.universal-property-dependent-pair-types
diff --git a/src/foundation/dependent-identifications.lagda.md b/src/foundation/dependent-identifications.lagda.md
index a194b8477e..50838bcafc 100644
--- a/src/foundation/dependent-identifications.lagda.md
+++ b/src/foundation/dependent-identifications.lagda.md
@@ -10,7 +10,7 @@ open import foundation-core.dependent-identifications public
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.equivalences
diff --git a/src/foundation/double-negation-modality.lagda.md b/src/foundation/double-negation-modality.lagda.md
index 8420266ab1..ee2407d3ea 100644
--- a/src/foundation/double-negation-modality.lagda.md
+++ b/src/foundation/double-negation-modality.lagda.md
@@ -12,7 +12,7 @@ open import foundation.universe-levels
 
 open import foundation-core.function-types
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 
 open import orthogonal-factorization-systems.local-types
 open import orthogonal-factorization-systems.modal-operators
diff --git a/src/foundation/equality-cartesian-product-types.lagda.md b/src/foundation/equality-cartesian-product-types.lagda.md
index 5a4a6c6137..7e385c0e60 100644
--- a/src/foundation/equality-cartesian-product-types.lagda.md
+++ b/src/foundation/equality-cartesian-product-types.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/equality-dependent-pair-types.lagda.md b/src/foundation/equality-dependent-pair-types.lagda.md
index bfab09dcbb..e5aa286fc1 100644
--- a/src/foundation/equality-dependent-pair-types.lagda.md
+++ b/src/foundation/equality-dependent-pair-types.lagda.md
@@ -13,7 +13,7 @@ open import foundation.action-on-identifications-dependent-functions
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-identifications
 open import foundation.dependent-pair-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.function-types
diff --git a/src/foundation/equality-fibers-of-maps.lagda.md b/src/foundation/equality-fibers-of-maps.lagda.md
index 1e45c6ebde..aed80bb948 100644
--- a/src/foundation/equality-fibers-of-maps.lagda.md
+++ b/src/foundation/equality-fibers-of-maps.lagda.md
@@ -10,7 +10,7 @@ module foundation.equality-fibers-of-maps where
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.equality-dependent-pair-types
diff --git a/src/foundation/equivalence-relations.lagda.md b/src/foundation/equivalence-relations.lagda.md
index 05bf364beb..b6c3391c06 100644
--- a/src/foundation/equivalence-relations.lagda.md
+++ b/src/foundation/equivalence-relations.lagda.md
@@ -39,7 +39,7 @@ open import foundation-core.identity-types
 open import foundation-core.logical-equivalences
 open import foundation-core.propositions
 open import foundation-core.sets
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/exclusive-disjunction.lagda.md b/src/foundation/exclusive-disjunction.lagda.md
index 3e1431893c..059f4e02c9 100644
--- a/src/foundation/exclusive-disjunction.lagda.md
+++ b/src/foundation/exclusive-disjunction.lagda.md
@@ -33,7 +33,7 @@ open import foundation-core.functoriality-dependent-function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.identity-types
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 
 open import univalent-combinatorics.2-element-types
 open import univalent-combinatorics.equality-finite-types
diff --git a/src/foundation/fiber-inclusions.lagda.md b/src/foundation/fiber-inclusions.lagda.md
index 290012dcd7..82de0e1e41 100644
--- a/src/foundation/fiber-inclusions.lagda.md
+++ b/src/foundation/fiber-inclusions.lagda.md
@@ -12,7 +12,7 @@ open import foundation.cones-over-cospans
 open import foundation.dependent-pair-types
 open import foundation.faithful-maps
 open import foundation.fibers-of-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/foundation/full-subtypes.lagda.md b/src/foundation/full-subtypes.lagda.md
index da27a78574..57a9714080 100644
--- a/src/foundation/full-subtypes.lagda.md
+++ b/src/foundation/full-subtypes.lagda.md
@@ -16,7 +16,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/function-types.lagda.md b/src/foundation/function-types.lagda.md
index a14e912eb0..cbafd7364a 100644
--- a/src/foundation/function-types.lagda.md
+++ b/src/foundation/function-types.lagda.md
@@ -16,7 +16,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/functoriality-coproduct-types.lagda.md b/src/foundation/functoriality-coproduct-types.lagda.md
index e534ac71ee..bb0ed02ec6 100644
--- a/src/foundation/functoriality-coproduct-types.lagda.md
+++ b/src/foundation/functoriality-coproduct-types.lagda.md
@@ -35,7 +35,7 @@ open import foundation-core.identity-types
 open import foundation-core.injective-maps
 open import foundation-core.negation
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/functoriality-dependent-function-types.lagda.md b/src/foundation/functoriality-dependent-function-types.lagda.md
index 7d0fb34e88..d627e7290b 100644
--- a/src/foundation/functoriality-dependent-function-types.lagda.md
+++ b/src/foundation/functoriality-dependent-function-types.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalence-extensionality
 open import foundation.equivalences
 open import foundation.function-extensionality
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/foundation/functoriality-dependent-pair-types.lagda.md b/src/foundation/functoriality-dependent-pair-types.lagda.md
index 595ee6e06d..f266b6f5bf 100644
--- a/src/foundation/functoriality-dependent-pair-types.lagda.md
+++ b/src/foundation/functoriality-dependent-pair-types.lagda.md
@@ -20,7 +20,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.pullbacks
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/homotopies.lagda.md b/src/foundation/homotopies.lagda.md
index 9ca7f628db..7dc0b3dd72 100644
--- a/src/foundation/homotopies.lagda.md
+++ b/src/foundation/homotopies.lagda.md
@@ -23,7 +23,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.sections
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/identity-systems.lagda.md b/src/foundation/identity-systems.lagda.md
index da6502913b..494a0cc6f6 100644
--- a/src/foundation/identity-systems.lagda.md
+++ b/src/foundation/identity-systems.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.contractible-types
 open import foundation-core.equivalences
 open import foundation-core.identity-types
 open import foundation-core.sections
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/identity-types.lagda.md b/src/foundation/identity-types.lagda.md
index f411f6e8d2..fd956e12c8 100644
--- a/src/foundation/identity-types.lagda.md
+++ b/src/foundation/identity-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
@@ -55,8 +55,8 @@ identifications in arbitrary types.
 | Path algebra                                     | [`foundation.path-algebra`](foundation.path-algebra.md)                                                                   |
 | Symmetric identity types                         | [`foundation.symmetric-identity-types`](foundation.symmetric-identity-types.md)                                           |
 | Torsorial type families                          | [`foundation.torsorial-type-families`](foundation.torsorial-type-families.md)                                             |
-| Transport (foundation)                           | [`foundation.transport`](foundation.transport.md)                                                                         |
-| Transport (foundation-core)                      | [`foundation-core.transport`](foundation-core.transport.md)                                                               |
+| Transport (foundation)                           | [`foundation.transport-along-identifications`](foundation.transport-along-identifications.md)                             |
+| Transport (foundation-core)                      | [`foundation-core.transport-along-identifications`](foundation-core.transport-along-identifications.md)                   |
 | The universal property of identity types         | [`foundation.universal-property-identity-types`](foundation.universal-property-identity-types.md)                         |
 
 ## Properties
diff --git a/src/foundation/induction-principle-propositional-truncation.lagda.md b/src/foundation/induction-principle-propositional-truncation.lagda.md
index 803ab7d6a0..dafe752efa 100644
--- a/src/foundation/induction-principle-propositional-truncation.lagda.md
+++ b/src/foundation/induction-principle-propositional-truncation.lagda.md
@@ -12,7 +12,7 @@ open import foundation.universe-levels
 
 open import foundation-core.identity-types
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/isolated-points.lagda.md b/src/foundation/isolated-points.lagda.md
index 0d12b9518e..e9196a4deb 100644
--- a/src/foundation/isolated-points.lagda.md
+++ b/src/foundation/isolated-points.lagda.md
@@ -35,7 +35,7 @@ open import foundation-core.injective-maps
 open import foundation-core.propositions
 open import foundation-core.sets
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/locally-small-types.lagda.md b/src/foundation/locally-small-types.lagda.md
index dfcfb2b8d9..f16321383d 100644
--- a/src/foundation/locally-small-types.lagda.md
+++ b/src/foundation/locally-small-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.small-types
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation/partitions.lagda.md b/src/foundation/partitions.lagda.md
index fa2671d2c3..b86e92a187 100644
--- a/src/foundation/partitions.lagda.md
+++ b/src/foundation/partitions.lagda.md
@@ -25,7 +25,7 @@ open import foundation.small-types
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
 open import foundation.surjective-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/foundation/perfect-images.lagda.md b/src/foundation/perfect-images.lagda.md
index 586955f0c0..d5e2a9986e 100644
--- a/src/foundation/perfect-images.lagda.md
+++ b/src/foundation/perfect-images.lagda.md
@@ -28,7 +28,7 @@ open import foundation-core.identity-types
 open import foundation-core.injective-maps
 open import foundation-core.propositional-maps
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/pointed-torsorial-type-families.lagda.md b/src/foundation/pointed-torsorial-type-families.lagda.md
index 968bfbf1ca..d8fcdc590e 100644
--- a/src/foundation/pointed-torsorial-type-families.lagda.md
+++ b/src/foundation/pointed-torsorial-type-families.lagda.md
@@ -14,7 +14,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.locally-small-types
 open import foundation.propositional-truncations
 open import foundation.sorial-type-families
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.universe-levels
 
diff --git a/src/foundation/products-unordered-pairs-of-types.lagda.md b/src/foundation/products-unordered-pairs-of-types.lagda.md
index 01b4238bfb..5741e3ab94 100644
--- a/src/foundation/products-unordered-pairs-of-types.lagda.md
+++ b/src/foundation/products-unordered-pairs-of-types.lagda.md
@@ -11,7 +11,7 @@ open import foundation.dependent-pair-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-dependent-function-types
 open import foundation.symmetric-operations
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 open import foundation.unordered-pairs
 open import foundation.unordered-pairs-of-types
diff --git a/src/foundation/propositional-extensionality.lagda.md b/src/foundation/propositional-extensionality.lagda.md
index f1c2fd96b1..0154bd5709 100644
--- a/src/foundation/propositional-extensionality.lagda.md
+++ b/src/foundation/propositional-extensionality.lagda.md
@@ -15,7 +15,7 @@ open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.raising-universe-levels
 open import foundation.subtype-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md
index 8342afce15..a202424801 100644
--- a/src/foundation/propositional-truncations.lagda.md
+++ b/src/foundation/propositional-truncations.lagda.md
@@ -23,7 +23,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.sets
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation/pullbacks.lagda.md b/src/foundation/pullbacks.lagda.md
index ace83663d3..6d4cec21c1 100644
--- a/src/foundation/pullbacks.lagda.md
+++ b/src/foundation/pullbacks.lagda.md
@@ -31,7 +31,7 @@ open import foundation-core.equality-dependent-pair-types
 open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-function-types
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/singleton-induction.lagda.md b/src/foundation/singleton-induction.lagda.md
index 79b42d767f..ed4b0a5879 100644
--- a/src/foundation/singleton-induction.lagda.md
+++ b/src/foundation/singleton-induction.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.sections
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/structure.lagda.md b/src/foundation/structure.lagda.md
index 605062f4e3..dea3740006 100644
--- a/src/foundation/structure.lagda.md
+++ b/src/foundation/structure.lagda.md
@@ -14,7 +14,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.fibers-of-maps
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/subuniverses.lagda.md b/src/foundation/subuniverses.lagda.md
index 97f5866997..5c1d434904 100644
--- a/src/foundation/subuniverses.lagda.md
+++ b/src/foundation/subuniverses.lagda.md
@@ -20,7 +20,7 @@ open import foundation-core.equivalences
 open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/symmetric-difference.lagda.md b/src/foundation/symmetric-difference.lagda.md
index 80f7687779..bf91677b74 100644
--- a/src/foundation/symmetric-difference.lagda.md
+++ b/src/foundation/symmetric-difference.lagda.md
@@ -21,7 +21,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index c646e8dfa2..fd45f6d1d1 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -13,7 +13,7 @@ open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
 open import foundation.equivalences
 open import foundation.fundamental-theorem-of-identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/foundation/transport.lagda.md b/src/foundation/transport-along-identifications.lagda.md
similarity index 85%
rename from src/foundation/transport.lagda.md
rename to src/foundation/transport-along-identifications.lagda.md
index 8320807035..9b917a57da 100644
--- a/src/foundation/transport.lagda.md
+++ b/src/foundation/transport-along-identifications.lagda.md
@@ -1,9 +1,9 @@
-# Transport
+# Transport along identifications
 
 ```agda
-module foundation.transport where
+module foundation.transport-along-identifications where
 
-open import foundation-core.transport public
+open import foundation-core.transport-along-identifications public
 ```
 
 <details><summary>Imports</summary>
@@ -25,8 +25,9 @@ open import foundation-core.identity-types
 
 Given a type family `B` over `A`, an
 [identification](foundation-core.identity-types.md) `p : x ＝ y` in `A` and an
-element `b : B x`, we can [**transport**](foundation-core.transport.md) the
-element `b` along the identification `p` to obtain an element `tr B p b : B y`.
+element `b : B x`, we can
+[**transport**](foundation-core.transport-along-identifications.md) the element
+`b` along the identification `p` to obtain an element `tr B p b : B y`.
 
 The fact that `tr B p` is an [equivalence](foundation-core.equivalences.md) is
 recorded in this file.
diff --git a/src/foundation/type-theoretic-principle-of-choice.lagda.md b/src/foundation/type-theoretic-principle-of-choice.lagda.md
index 1a1a7ef294..65edfc98a1 100644
--- a/src/foundation/type-theoretic-principle-of-choice.lagda.md
+++ b/src/foundation/type-theoretic-principle-of-choice.lagda.md
@@ -17,7 +17,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/univalence-action-on-equivalences.lagda.md b/src/foundation/univalence-action-on-equivalences.lagda.md
index cbf0b36566..6eaf2eafcb 100644
--- a/src/foundation/univalence-action-on-equivalences.lagda.md
+++ b/src/foundation/univalence-action-on-equivalences.lagda.md
@@ -12,7 +12,7 @@ open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.sets
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/foundation/univalence-implies-function-extensionality.lagda.md b/src/foundation/univalence-implies-function-extensionality.lagda.md
index 87a27e5e83..b374530cb1 100644
--- a/src/foundation/univalence-implies-function-extensionality.lagda.md
+++ b/src/foundation/univalence-implies-function-extensionality.lagda.md
@@ -20,7 +20,7 @@ open import foundation-core.function-extensionality
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/univalent-type-families.lagda.md b/src/foundation/univalent-type-families.lagda.md
index 5d2e7960cc..8297c1c7a7 100644
--- a/src/foundation/univalent-type-families.lagda.md
+++ b/src/foundation/univalent-type-families.lagda.md
@@ -7,7 +7,7 @@ module foundation.univalent-type-families where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.equivalences
diff --git a/src/foundation/universal-property-image.lagda.md b/src/foundation/universal-property-image.lagda.md
index 3754713693..1dc5d76fd3 100644
--- a/src/foundation/universal-property-image.lagda.md
+++ b/src/foundation/universal-property-image.lagda.md
@@ -13,7 +13,7 @@ open import foundation.images
 open import foundation.propositional-truncations
 open import foundation.slice
 open import foundation.surjective-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.contractible-maps
diff --git a/src/foundation/universal-property-set-quotients.lagda.md b/src/foundation/universal-property-set-quotients.lagda.md
index 203fa60492..1ff24df007 100644
--- a/src/foundation/universal-property-set-quotients.lagda.md
+++ b/src/foundation/universal-property-set-quotients.lagda.md
@@ -25,7 +25,7 @@ open import foundation.propositional-truncations
 open import foundation.reflecting-maps-equivalence-relations
 open import foundation.sets
 open import foundation.surjective-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universal-property-image
 open import foundation.universe-levels
 
diff --git a/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md b/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md
index 309e845787..74009d5e07 100644
--- a/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md
+++ b/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md
@@ -17,7 +17,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.enriched-undirected-graphs
diff --git a/src/graph-theory/equivalences-undirected-graphs.lagda.md b/src/graph-theory/equivalences-undirected-graphs.lagda.md
index 050660708e..958f6e24b6 100644
--- a/src/graph-theory/equivalences-undirected-graphs.lagda.md
+++ b/src/graph-theory/equivalences-undirected-graphs.lagda.md
@@ -20,7 +20,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 open import foundation.unordered-pairs
diff --git a/src/graph-theory/fibers-directed-graphs.lagda.md b/src/graph-theory/fibers-directed-graphs.lagda.md
index 2325a19107..3b7917dec1 100644
--- a/src/graph-theory/fibers-directed-graphs.lagda.md
+++ b/src/graph-theory/fibers-directed-graphs.lagda.md
@@ -14,7 +14,7 @@ open import foundation.equality-dependent-pair-types
 open import foundation.equivalences
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/graph-theory/morphisms-undirected-graphs.lagda.md b/src/graph-theory/morphisms-undirected-graphs.lagda.md
index de1a35bfe2..004e717c9b 100644
--- a/src/graph-theory/morphisms-undirected-graphs.lagda.md
+++ b/src/graph-theory/morphisms-undirected-graphs.lagda.md
@@ -19,7 +19,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 open import foundation.unordered-pairs
 
diff --git a/src/graph-theory/undirected-graphs.lagda.md b/src/graph-theory/undirected-graphs.lagda.md
index cab82a2c23..f7b5cbd8f5 100644
--- a/src/graph-theory/undirected-graphs.lagda.md
+++ b/src/graph-theory/undirected-graphs.lagda.md
@@ -10,7 +10,7 @@ module graph-theory.undirected-graphs where
 open import foundation.cartesian-product-types
 open import foundation.dependent-pair-types
 open import foundation.equivalences
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 open import foundation.unordered-pairs
 
diff --git a/src/graph-theory/walks-directed-graphs.lagda.md b/src/graph-theory/walks-directed-graphs.lagda.md
index cb327fdab1..0b38186d0a 100644
--- a/src/graph-theory/walks-directed-graphs.lagda.md
+++ b/src/graph-theory/walks-directed-graphs.lagda.md
@@ -24,7 +24,7 @@ open import foundation.identity-types
 open import foundation.injective-maps
 open import foundation.negation
 open import foundation.raising-universe-levels
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.directed-graphs
diff --git a/src/group-theory/concrete-group-actions.lagda.md b/src/group-theory/concrete-group-actions.lagda.md
index bc5108da26..fe407ed17e 100644
--- a/src/group-theory/concrete-group-actions.lagda.md
+++ b/src/group-theory/concrete-group-actions.lagda.md
@@ -9,7 +9,7 @@ module group-theory.concrete-group-actions where
 ```agda
 open import foundation.function-types
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.concrete-groups
diff --git a/src/group-theory/homomorphisms-concrete-group-actions.lagda.md b/src/group-theory/homomorphisms-concrete-group-actions.lagda.md
index b86eae7b09..62057b8e87 100644
--- a/src/group-theory/homomorphisms-concrete-group-actions.lagda.md
+++ b/src/group-theory/homomorphisms-concrete-group-actions.lagda.md
@@ -16,7 +16,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.concrete-group-actions
diff --git a/src/group-theory/normal-subgroups.lagda.md b/src/group-theory/normal-subgroups.lagda.md
index 36448d9058..eadcaf09fb 100644
--- a/src/group-theory/normal-subgroups.lagda.md
+++ b/src/group-theory/normal-subgroups.lagda.md
@@ -20,7 +20,7 @@ open import foundation.large-binary-relations
 open import foundation.propositions
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.congruence-relations-groups
diff --git a/src/group-theory/subgroups.lagda.md b/src/group-theory/subgroups.lagda.md
index a14391c6d1..beed392e57 100644
--- a/src/group-theory/subgroups.lagda.md
+++ b/src/group-theory/subgroups.lagda.md
@@ -25,7 +25,7 @@ open import foundation.propositions
 open import foundation.sets
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.groups
diff --git a/src/group-theory/torsors.lagda.md b/src/group-theory/torsors.lagda.md
index 5759ad386f..4a2f87f322 100644
--- a/src/group-theory/torsors.lagda.md
+++ b/src/group-theory/torsors.lagda.md
@@ -21,7 +21,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtype-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.concrete-groups
diff --git a/src/group-theory/transitive-concrete-group-actions.lagda.md b/src/group-theory/transitive-concrete-group-actions.lagda.md
index dd3eac590f..2d589c0391 100644
--- a/src/group-theory/transitive-concrete-group-actions.lagda.md
+++ b/src/group-theory/transitive-concrete-group-actions.lagda.md
@@ -21,7 +21,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.concrete-group-actions
diff --git a/src/higher-group-theory/higher-group-actions.lagda.md b/src/higher-group-theory/higher-group-actions.lagda.md
index 3e339c1a9f..92a3590986 100644
--- a/src/higher-group-theory/higher-group-actions.lagda.md
+++ b/src/higher-group-theory/higher-group-actions.lagda.md
@@ -8,7 +8,7 @@ module higher-group-theory.higher-group-actions where
 
 ```agda
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import higher-group-theory.higher-groups
diff --git a/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md b/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md
index 6d2ee91be6..5929ab0dbc 100644
--- a/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md
+++ b/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md
@@ -12,7 +12,7 @@ open import foundation.equivalences
 open import foundation.function-extensionality
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import higher-group-theory.higher-group-actions
diff --git a/src/linear-algebra/vectors.lagda.md b/src/linear-algebra/vectors.lagda.md
index 2fb22bb53a..4547743267 100644
--- a/src/linear-algebra/vectors.lagda.md
+++ b/src/linear-algebra/vectors.lagda.md
@@ -21,7 +21,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.unit-type
diff --git a/src/lists/functoriality-lists.lagda.md b/src/lists/functoriality-lists.lagda.md
index 62b9b8f9c9..fe70813734 100644
--- a/src/lists/functoriality-lists.lagda.md
+++ b/src/lists/functoriality-lists.lagda.md
@@ -16,7 +16,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import linear-algebra.functoriality-vectors
diff --git a/src/lists/permutation-lists.lagda.md b/src/lists/permutation-lists.lagda.md
index cab237461e..b76eba9548 100644
--- a/src/lists/permutation-lists.lagda.md
+++ b/src/lists/permutation-lists.lagda.md
@@ -16,7 +16,7 @@ open import foundation.dependent-pair-types
 open import foundation.equality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import linear-algebra.functoriality-vectors
diff --git a/src/lists/permutation-vectors.lagda.md b/src/lists/permutation-vectors.lagda.md
index 16572742fd..ae8d72e68a 100644
--- a/src/lists/permutation-vectors.lagda.md
+++ b/src/lists/permutation-vectors.lagda.md
@@ -22,7 +22,7 @@ open import foundation.function-types
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import linear-algebra.functoriality-vectors
diff --git a/src/lists/sort-by-insertion-vectors.lagda.md b/src/lists/sort-by-insertion-vectors.lagda.md
index 2f9c6c74e3..6f756f9386 100644
--- a/src/lists/sort-by-insertion-vectors.lagda.md
+++ b/src/lists/sort-by-insertion-vectors.lagda.md
@@ -18,7 +18,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/order-theory/finitely-graded-posets.lagda.md b/src/order-theory/finitely-graded-posets.lagda.md
index 2b684e743a..7534d942df 100644
--- a/src/order-theory/finitely-graded-posets.lagda.md
+++ b/src/order-theory/finitely-graded-posets.lagda.md
@@ -24,7 +24,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/organic-chemistry/ethane.lagda.md b/src/organic-chemistry/ethane.lagda.md
index 74afce7451..a284da7eff 100644
--- a/src/organic-chemistry/ethane.lagda.md
+++ b/src/organic-chemistry/ethane.lagda.md
@@ -23,7 +23,7 @@ open import foundation.identity-types
 open import foundation.injective-maps
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 open import foundation.unordered-pairs
diff --git a/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md b/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md
index 205589c532..3d30a6cbbc 100644
--- a/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md
+++ b/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md
@@ -26,7 +26,7 @@ open import foundation.monomorphisms
 open import foundation.propositions
 open import foundation.sets
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-arithmetic-dependent-pair-types
diff --git a/src/orthogonal-factorization-systems/open-modalities.lagda.md b/src/orthogonal-factorization-systems/open-modalities.lagda.md
index c716ec3631..6268816a97 100644
--- a/src/orthogonal-factorization-systems/open-modalities.lagda.md
+++ b/src/orthogonal-factorization-systems/open-modalities.lagda.md
@@ -14,7 +14,7 @@ open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.locally-small-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import orthogonal-factorization-systems.higher-modalities
diff --git a/src/polytopes/abstract-polytopes.lagda.md b/src/polytopes/abstract-polytopes.lagda.md
index 02b69a6446..a490cdeffd 100644
--- a/src/polytopes/abstract-polytopes.lagda.md
+++ b/src/polytopes/abstract-polytopes.lagda.md
@@ -17,7 +17,7 @@ open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md
index e2f8fdf80b..c458e6b9a8 100644
--- a/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md
+++ b/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md
@@ -17,7 +17,7 @@ open import foundation.logical-equivalences
 open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unions-subtypes
 open import foundation.universe-levels
 
diff --git a/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md
index a10777ae7b..3e3e087bfe 100644
--- a/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md
+++ b/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md
@@ -18,7 +18,7 @@ open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unions-subtypes
 open import foundation.universe-levels
 
diff --git a/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md
index 8e268bd112..cec34c1f4e 100644
--- a/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md
+++ b/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md
@@ -18,7 +18,7 @@ open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unions-subtypes
 open import foundation.universe-levels
 
diff --git a/src/set-theory/cumulative-hierarchy.lagda.md b/src/set-theory/cumulative-hierarchy.lagda.md
index a499a76770..6ce0932a89 100644
--- a/src/set-theory/cumulative-hierarchy.lagda.md
+++ b/src/set-theory/cumulative-hierarchy.lagda.md
@@ -27,7 +27,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md
index ee599d6e21..1611c57132 100644
--- a/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md
@@ -22,7 +22,7 @@ open import foundation.relaxed-sigma-decompositions
 open import foundation.sigma-closed-subuniverses
 open import foundation.sigma-decomposition-subuniverse
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-theoretic-principle-of-choice
diff --git a/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md
index 050d359dd3..4479561277 100644
--- a/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md
@@ -22,7 +22,7 @@ open import foundation.relaxed-sigma-decompositions
 open import foundation.sigma-closed-subuniverses
 open import foundation.sigma-decomposition-subuniverse
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md
index 50ee887c03..8a891eacda 100644
--- a/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md
@@ -20,7 +20,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
diff --git a/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md b/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md
index e7b28ec46c..025f737803 100644
--- a/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md
@@ -20,7 +20,7 @@ open import foundation.pi-decompositions-subuniverse
 open import foundation.product-decompositions
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.univalence
 open import foundation.universe-levels
diff --git a/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md b/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md
index 4fedc029c4..2518d8c0bc 100644
--- a/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md
@@ -19,7 +19,7 @@ open import foundation.product-decompositions
 open import foundation.product-decompositions-subuniverse
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
diff --git a/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md b/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md
index 16a59ec9f6..ff5a2545b2 100644
--- a/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md
@@ -22,7 +22,7 @@ open import foundation.sigma-closed-subuniverses
 open import foundation.sigma-decomposition-subuniverse
 open import foundation.small-types
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-theoretic-principle-of-choice
diff --git a/src/species/species-of-types-in-subuniverses.lagda.md b/src/species/species-of-types-in-subuniverses.lagda.md
index 890975d154..1ef1c85fec 100644
--- a/src/species/species-of-types-in-subuniverses.lagda.md
+++ b/src/species/species-of-types-in-subuniverses.lagda.md
@@ -13,7 +13,7 @@ open import foundation.equivalences
 open import foundation.function-types
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/species/species-of-types.lagda.md b/src/species/species-of-types.lagda.md
index 585fc15a8c..ceb687541f 100644
--- a/src/species/species-of-types.lagda.md
+++ b/src/species/species-of-types.lagda.md
@@ -9,7 +9,7 @@ module species.species-of-types where
 ```agda
 open import foundation.cartesian-product-types
 open import foundation.equivalences
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 ```
diff --git a/src/structured-types/pointed-maps.lagda.md b/src/structured-types/pointed-maps.lagda.md
index 0cdb448f97..ff004374cb 100644
--- a/src/structured-types/pointed-maps.lagda.md
+++ b/src/structured-types/pointed-maps.lagda.md
@@ -13,7 +13,7 @@ open import foundation.constant-maps
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import structured-types.pointed-dependent-functions
diff --git a/src/synthetic-homotopy-theory/26-descent.lagda.md b/src/synthetic-homotopy-theory/26-descent.lagda.md
index 1535e6ce6c..713d873a48 100644
--- a/src/synthetic-homotopy-theory/26-descent.lagda.md
+++ b/src/synthetic-homotopy-theory/26-descent.lagda.md
@@ -30,7 +30,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.pullbacks
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.univalence
 open import foundation.universe-levels
diff --git a/src/synthetic-homotopy-theory/26-id-pushout.lagda.md b/src/synthetic-homotopy-theory/26-id-pushout.lagda.md
index 955cbf3c7b..4163a33a94 100644
--- a/src/synthetic-homotopy-theory/26-id-pushout.lagda.md
+++ b/src/synthetic-homotopy-theory/26-id-pushout.lagda.md
@@ -22,7 +22,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universal-property-identity-types
 open import foundation.universe-levels
 
diff --git a/src/synthetic-homotopy-theory/circle.lagda.md b/src/synthetic-homotopy-theory/circle.lagda.md
index b33a747e72..71067c794a 100644
--- a/src/synthetic-homotopy-theory/circle.lagda.md
+++ b/src/synthetic-homotopy-theory/circle.lagda.md
@@ -16,7 +16,7 @@ open import foundation.identity-types
 open import foundation.mere-equality
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import structured-types.pointed-types
diff --git a/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md b/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md
index 1c6f3f5ea4..fd04ca5e1f 100644
--- a/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md
@@ -19,7 +19,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.cocones-under-spans
diff --git a/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md b/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md
index 96775f12df..2017650ebe 100644
--- a/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md
@@ -12,7 +12,7 @@ open import foundation.cones-over-cospans
 open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.pullbacks
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.cocones-under-spans
diff --git a/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md b/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md
index df4f68177f..b4e7a6a56e 100644
--- a/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md
@@ -20,7 +20,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.pullbacks
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.cocones-under-spans
diff --git a/src/synthetic-homotopy-theory/descent-circle.lagda.md b/src/synthetic-homotopy-theory/descent-circle.lagda.md
index 1efe950f82..79f5f51d8e 100644
--- a/src/synthetic-homotopy-theory/descent-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/descent-circle.lagda.md
@@ -28,7 +28,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.path-algebra
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/synthetic-homotopy-theory/free-loops.lagda.md b/src/synthetic-homotopy-theory/free-loops.lagda.md
index 5e83bb9173..9fe6f480e3 100644
--- a/src/synthetic-homotopy-theory/free-loops.lagda.md
+++ b/src/synthetic-homotopy-theory/free-loops.lagda.md
@@ -16,7 +16,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/synthetic-homotopy-theory/interval-type.lagda.md b/src/synthetic-homotopy-theory/interval-type.lagda.md
index 5013094d8e..6174817856 100644
--- a/src/synthetic-homotopy-theory/interval-type.lagda.md
+++ b/src/synthetic-homotopy-theory/interval-type.lagda.md
@@ -15,7 +15,7 @@ open import foundation.equivalences
 open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/synthetic-homotopy-theory/multiplication-circle.lagda.md b/src/synthetic-homotopy-theory/multiplication-circle.lagda.md
index 4dbcf962ac..22827748bd 100644
--- a/src/synthetic-homotopy-theory/multiplication-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/multiplication-circle.lagda.md
@@ -13,7 +13,7 @@ open import foundation.function-extensionality
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import structured-types.pointed-homotopies
diff --git a/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md b/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md
index 18bfcc64bb..15f2fe35d8 100644
--- a/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md
+++ b/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md
index 5802758f3c..0f77607fe0 100644
--- a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md
@@ -30,7 +30,7 @@ open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.sets
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.universe-levels
diff --git a/src/synthetic-homotopy-theory/universal-property-circle.lagda.md b/src/synthetic-homotopy-theory/universal-property-circle.lagda.md
index 002fc6ebb1..9945df1d1f 100644
--- a/src/synthetic-homotopy-theory/universal-property-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-property-circle.lagda.md
@@ -22,7 +22,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sections
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.free-loops
diff --git a/src/trees/bases-directed-trees.lagda.md b/src/trees/bases-directed-trees.lagda.md
index fa49199117..62493b1ade 100644
--- a/src/trees/bases-directed-trees.lagda.md
+++ b/src/trees/bases-directed-trees.lagda.md
@@ -21,7 +21,7 @@ open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
 open import foundation.universe-levels
diff --git a/src/trees/directed-trees.lagda.md b/src/trees/directed-trees.lagda.md
index 8c72d592d4..629fb4c5bb 100644
--- a/src/trees/directed-trees.lagda.md
+++ b/src/trees/directed-trees.lagda.md
@@ -25,7 +25,7 @@ open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
diff --git a/src/trees/elementhood-relation-w-types.lagda.md b/src/trees/elementhood-relation-w-types.lagda.md
index 342b795a11..2e980645a5 100644
--- a/src/trees/elementhood-relation-w-types.lagda.md
+++ b/src/trees/elementhood-relation-w-types.lagda.md
@@ -10,7 +10,7 @@ module trees.elementhood-relation-w-types where
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.elementhood-relation-coalgebras-polynomial-endofunctors
diff --git a/src/trees/enriched-directed-trees.lagda.md b/src/trees/enriched-directed-trees.lagda.md
index 57fbf3744d..570286771d 100644
--- a/src/trees/enriched-directed-trees.lagda.md
+++ b/src/trees/enriched-directed-trees.lagda.md
@@ -20,7 +20,7 @@ open import foundation.identity-types
 open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.directed-graphs
diff --git a/src/trees/equivalences-directed-trees.lagda.md b/src/trees/equivalences-directed-trees.lagda.md
index 89badad851..c508da1874 100644
--- a/src/trees/equivalences-directed-trees.lagda.md
+++ b/src/trees/equivalences-directed-trees.lagda.md
@@ -19,7 +19,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.subtype-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.equivalences-directed-graphs
diff --git a/src/trees/equivalences-enriched-directed-trees.lagda.md b/src/trees/equivalences-enriched-directed-trees.lagda.md
index e24e0e65d7..c796443d76 100644
--- a/src/trees/equivalences-enriched-directed-trees.lagda.md
+++ b/src/trees/equivalences-enriched-directed-trees.lagda.md
@@ -20,7 +20,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.enriched-directed-trees
diff --git a/src/trees/extensional-w-types.lagda.md b/src/trees/extensional-w-types.lagda.md
index d53f7b4c1a..3a8887dc5d 100644
--- a/src/trees/extensional-w-types.lagda.md
+++ b/src/trees/extensional-w-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.slice
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.univalent-type-families
 open import foundation.universe-levels
diff --git a/src/trees/functoriality-combinator-directed-trees.lagda.md b/src/trees/functoriality-combinator-directed-trees.lagda.md
index b181907762..62ecdf2b9c 100644
--- a/src/trees/functoriality-combinator-directed-trees.lagda.md
+++ b/src/trees/functoriality-combinator-directed-trees.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.combinator-directed-trees
diff --git a/src/trees/functoriality-w-types.lagda.md b/src/trees/functoriality-w-types.lagda.md
index f55e1f0957..51e33d255b 100644
--- a/src/trees/functoriality-w-types.lagda.md
+++ b/src/trees/functoriality-w-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.functoriality-dependent-function-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositional-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-maps
 open import foundation.truncated-types
 open import foundation.truncation-levels
diff --git a/src/trees/induction-w-types.lagda.md b/src/trees/induction-w-types.lagda.md
index bf924b6e13..1805b55311 100644
--- a/src/trees/induction-w-types.lagda.md
+++ b/src/trees/induction-w-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation.function-extensionality
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.elementhood-relation-w-types
diff --git a/src/trees/morphisms-directed-trees.lagda.md b/src/trees/morphisms-directed-trees.lagda.md
index f4cb1d987a..eea276a027 100644
--- a/src/trees/morphisms-directed-trees.lagda.md
+++ b/src/trees/morphisms-directed-trees.lagda.md
@@ -17,7 +17,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-empty-type
 open import foundation.universe-levels
 
diff --git a/src/trees/morphisms-enriched-directed-trees.lagda.md b/src/trees/morphisms-enriched-directed-trees.lagda.md
index ea3167e9b1..c6618ae770 100644
--- a/src/trees/morphisms-enriched-directed-trees.lagda.md
+++ b/src/trees/morphisms-enriched-directed-trees.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.enriched-directed-trees
diff --git a/src/trees/polynomial-endofunctors.lagda.md b/src/trees/polynomial-endofunctors.lagda.md
index 96be64542a..d2d8d3280d 100644
--- a/src/trees/polynomial-endofunctors.lagda.md
+++ b/src/trees/polynomial-endofunctors.lagda.md
@@ -16,7 +16,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/trees/ranks-of-elements-w-types.lagda.md b/src/trees/ranks-of-elements-w-types.lagda.md
index 4ef53cb626..1efe2269d1 100644
--- a/src/trees/ranks-of-elements-w-types.lagda.md
+++ b/src/trees/ranks-of-elements-w-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.elementhood-relation-w-types
diff --git a/src/trees/rooted-morphisms-directed-trees.lagda.md b/src/trees/rooted-morphisms-directed-trees.lagda.md
index 2b46217d61..aad2dd69aa 100644
--- a/src/trees/rooted-morphisms-directed-trees.lagda.md
+++ b/src/trees/rooted-morphisms-directed-trees.lagda.md
@@ -18,7 +18,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subtype-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.bases-directed-trees
diff --git a/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md b/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md
index 1836a44f6d..f876ff518a 100644
--- a/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md
+++ b/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md
@@ -12,7 +12,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.enriched-directed-trees
diff --git a/src/trees/small-multisets.lagda.md b/src/trees/small-multisets.lagda.md
index 63341f8813..acfd2d4d2c 100644
--- a/src/trees/small-multisets.lagda.md
+++ b/src/trees/small-multisets.lagda.md
@@ -18,7 +18,7 @@ open import foundation.identity-types
 open import foundation.propositions
 open import foundation.small-types
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md b/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md
index 5a420b099b..51c5d408dd 100644
--- a/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md
+++ b/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md
@@ -23,7 +23,7 @@ open import foundation.identity-types
 open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-empty-type
 open import foundation.universe-levels
 
diff --git a/src/trees/underlying-trees-of-elements-of-w-types.lagda.md b/src/trees/underlying-trees-of-elements-of-w-types.lagda.md
index 2a196afe79..6579f766cd 100644
--- a/src/trees/underlying-trees-of-elements-of-w-types.lagda.md
+++ b/src/trees/underlying-trees-of-elements-of-w-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.identity-types
 open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.directed-graphs
diff --git a/src/trees/undirected-trees.lagda.md b/src/trees/undirected-trees.lagda.md
index 539065d90d..b9317cfec2 100644
--- a/src/trees/undirected-trees.lagda.md
+++ b/src/trees/undirected-trees.lagda.md
@@ -20,7 +20,7 @@ open import foundation.mere-equality
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.paths-undirected-graphs
diff --git a/src/trees/universal-multiset.lagda.md b/src/trees/universal-multiset.lagda.md
index 126ca1365c..274f1dfe83 100644
--- a/src/trees/universal-multiset.lagda.md
+++ b/src/trees/universal-multiset.lagda.md
@@ -13,7 +13,7 @@ open import foundation.identity-types
 open import foundation.raising-universe-levels
 open import foundation.small-types
 open import foundation.small-universes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.functoriality-w-types
diff --git a/src/trees/w-types.lagda.md b/src/trees/w-types.lagda.md
index 54a11efe7d..e8a3bd5de9 100644
--- a/src/trees/w-types.lagda.md
+++ b/src/trees/w-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-theoretic-principle-of-choice
diff --git a/src/type-theories/dependent-type-theories.lagda.md b/src/type-theories/dependent-type-theories.lagda.md
index 9fb662749e..3c9f43e8b1 100644
--- a/src/type-theories/dependent-type-theories.lagda.md
+++ b/src/type-theories/dependent-type-theories.lagda.md
@@ -17,7 +17,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/type-theories/fibered-dependent-type-theories.lagda.md b/src/type-theories/fibered-dependent-type-theories.lagda.md
index fef7eb8f07..a5da36d53a 100644
--- a/src/type-theories/fibered-dependent-type-theories.lagda.md
+++ b/src/type-theories/fibered-dependent-type-theories.lagda.md
@@ -13,7 +13,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import type-theories.dependent-type-theories
diff --git a/src/type-theories/sections-dependent-type-theories.lagda.md b/src/type-theories/sections-dependent-type-theories.lagda.md
index 57a5c18782..339dc4c6e7 100644
--- a/src/type-theories/sections-dependent-type-theories.lagda.md
+++ b/src/type-theories/sections-dependent-type-theories.lagda.md
@@ -10,7 +10,7 @@ module type-theories.sections-dependent-type-theories where
 
 ```agda
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import type-theories.dependent-type-theories
diff --git a/src/type-theories/simple-type-theories.lagda.md b/src/type-theories/simple-type-theories.lagda.md
index 227dc224ff..98461a1a5e 100644
--- a/src/type-theories/simple-type-theories.lagda.md
+++ b/src/type-theories/simple-type-theories.lagda.md
@@ -13,7 +13,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md
index 4f925227c3..335deed2c7 100644
--- a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md
+++ b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md
@@ -34,7 +34,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.univalence
 open import foundation.universe-levels
diff --git a/src/univalent-combinatorics/2-element-subtypes.lagda.md b/src/univalent-combinatorics/2-element-subtypes.lagda.md
index 5ba14b8c22..d41430d2f4 100644
--- a/src/univalent-combinatorics/2-element-subtypes.lagda.md
+++ b/src/univalent-combinatorics/2-element-subtypes.lagda.md
@@ -24,7 +24,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/univalent-combinatorics/2-element-types.lagda.md b/src/univalent-combinatorics/2-element-types.lagda.md
index 19d458a7af..ed91e496bd 100644
--- a/src/univalent-combinatorics/2-element-types.lagda.md
+++ b/src/univalent-combinatorics/2-element-types.lagda.md
@@ -37,7 +37,7 @@ open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.sets
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
diff --git a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md
index db7c453f7b..6e970321cd 100644
--- a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md
+++ b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md
@@ -26,7 +26,7 @@ open import foundation.functoriality-coproduct-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.sections
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-unit-type
diff --git a/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md b/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md
index f916a53857..a93706de19 100644
--- a/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md
+++ b/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md
@@ -22,7 +22,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.iterated-cartesian-product-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/univalent-combinatorics/decidable-subtypes.lagda.md b/src/univalent-combinatorics/decidable-subtypes.lagda.md
index 90a64f18aa..66b79ec69f 100644
--- a/src/univalent-combinatorics/decidable-subtypes.lagda.md
+++ b/src/univalent-combinatorics/decidable-subtypes.lagda.md
@@ -22,7 +22,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import univalent-combinatorics.decidable-dependent-pair-types
diff --git a/src/univalent-combinatorics/dependent-pair-types.lagda.md b/src/univalent-combinatorics/dependent-pair-types.lagda.md
index d7acd5e852..5ab79d007e 100644
--- a/src/univalent-combinatorics/dependent-pair-types.lagda.md
+++ b/src/univalent-combinatorics/dependent-pair-types.lagda.md
@@ -25,7 +25,7 @@ open import foundation.propositions
 open import foundation.sections
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
diff --git a/src/univalent-combinatorics/equivalences-cubes.lagda.md b/src/univalent-combinatorics/equivalences-cubes.lagda.md
index 89f968dd7a..f71497fe95 100644
--- a/src/univalent-combinatorics/equivalences-cubes.lagda.md
+++ b/src/univalent-combinatorics/equivalences-cubes.lagda.md
@@ -19,7 +19,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import univalent-combinatorics.cubes
diff --git a/src/univalent-combinatorics/fibers-of-maps.lagda.md b/src/univalent-combinatorics/fibers-of-maps.lagda.md
index 79a868aaca..7ad1cdbb42 100644
--- a/src/univalent-combinatorics/fibers-of-maps.lagda.md
+++ b/src/univalent-combinatorics/fibers-of-maps.lagda.md
@@ -21,7 +21,7 @@ open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sections
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/univalent-combinatorics/finite-types.lagda.md b/src/univalent-combinatorics/finite-types.lagda.md
index 2791564474..4fa2bac999 100644
--- a/src/univalent-combinatorics/finite-types.lagda.md
+++ b/src/univalent-combinatorics/finite-types.lagda.md
@@ -32,7 +32,7 @@ open import foundation.raising-universe-levels
 open import foundation.sets
 open import foundation.subtypes
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md
index 8dc231e357..9f1ddb69ab 100644
--- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md
+++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md
@@ -48,7 +48,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/univalent-combinatorics/pi-finite-types.lagda.md b/src/univalent-combinatorics/pi-finite-types.lagda.md
index d1adf3e29c..fb068418ad 100644
--- a/src/univalent-combinatorics/pi-finite-types.lagda.md
+++ b/src/univalent-combinatorics/pi-finite-types.lagda.md
@@ -46,7 +46,7 @@ open import foundation.set-truncations
 open import foundation.sets
 open import foundation.subtypes
 open import foundation.surjective-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-arithmetic-coproduct-types
diff --git a/src/universal-algebra/terms-over-signatures.lagda.md b/src/universal-algebra/terms-over-signatures.lagda.md
index 48f19c048d..8a5cf6febf 100644
--- a/src/universal-algebra/terms-over-signatures.lagda.md
+++ b/src/universal-algebra/terms-over-signatures.lagda.md
@@ -14,7 +14,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 

From de495cee0b5e26cf7be879df5868d43cd4cc31b1 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 15:04:05 +0200
Subject: [PATCH 09/37] "Transport along identifications" road map

---
 src/foundation-core/identity-types.lagda.md | 42 ++++++++++-----------
 src/foundation/identity-types.lagda.md      | 42 ++++++++++-----------
 2 files changed, 42 insertions(+), 42 deletions(-)

diff --git a/src/foundation-core/identity-types.lagda.md b/src/foundation-core/identity-types.lagda.md
index be603bd50c..1d93f9d7c0 100644
--- a/src/foundation-core/identity-types.lagda.md
+++ b/src/foundation-core/identity-types.lagda.md
@@ -49,27 +49,27 @@ width equals sign `＝`.
 The following table lists files that are about identity types and operations on
 identifications in arbitrary types.
 
-| Concept                                          | File                                                                                                                      |
-| ------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------- |
-| Action on identifications of binary functions    | [`foundation.action-on-identifications-binary-functions`](foundation.action-on-identifications-binary-functions.md)       |
-| Action on identifications of dependent functions | [`foundation.action-on-identifications-dependent-functions`](foundation.action-on-identifications-dependent-functions.md) |
-| Action on identifications of functions           | [`foundation.action-on-identifications-functions`](foundation.action-on-identifications-functions.md)                     |
-| Binary transport                                 | [`foundation.binary-transport`](foundation.binary-transport.md)                                                           |
-| Commuting squares of identifications             | [`foundation.commuting-squares-of-identifications`](foundation.commuting-squares-of-identifications.md)                   |
-| Dependent identifications (foundation)           | [`foundation.dependent-identifications`](foundation.dependent-identifications.md)                                         |
-| Dependent identifications (foundation-core)      | [`foundation-core.dependent-identifications`](foundation-core.dependent-identifications.md)                               |
-| The fundamental theorem of identity types        | [`foundation.fundamental-theorem-of-identity-types`](foundation.fundamental-theorem-of-identity-types.md)                 |
-| Hexagons of identifications                      | [`foundation.hexagons-of-identifications`](foundation.hexagons-of-identifications.md)                                     |
-| Identity systems                                 | [`foundation.identity-systems`](foundation.identity-systems.md)                                                           |
-| The identity type (foundation)                   | [`foundation.identity-types`](foundation.identity-types.md)                                                               |
-| The identity type (foundation-core)              | [`foundation-core.identity-types`](foundation-core.identity-types.md)                                                     |
-| Large identity types                             | [`foundation.large-identity-types`](foundation.large-identity-types.md)                                                   |
-| Path algebra                                     | [`foundation.path-algebra`](foundation.path-algebra.md)                                                                   |
-| Symmetric identity types                         | [`foundation.symmetric-identity-types`](foundation.symmetric-identity-types.md)                                           |
-| Torsorial type families                          | [`foundation.torsorial-type-families`](foundation.torsorial-type-families.md)                                             |
-| Transport (foundation)                           | [`foundation.transport-along-identifications`](foundation.transport-along-identifications.md)                             |
-| Transport (foundation-core)                      | [`foundation-core.transport-along-identifications`](foundation-core.transport-along-identifications.md)                   |
-| The universal property of identity types         | [`foundation.universal-property-identity-types`](foundation.universal-property-identity-types.md)                         |
+| Concept                                           | File                                                                                                                      |
+| ------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------- |
+| Action on identifications of binary functions     | [`foundation.action-on-identifications-binary-functions`](foundation.action-on-identifications-binary-functions.md)       |
+| Action on identifications of dependent functions  | [`foundation.action-on-identifications-dependent-functions`](foundation.action-on-identifications-dependent-functions.md) |
+| Action on identifications of functions            | [`foundation.action-on-identifications-functions`](foundation.action-on-identifications-functions.md)                     |
+| Binary transport                                  | [`foundation.binary-transport`](foundation.binary-transport.md)                                                           |
+| Commuting squares of identifications              | [`foundation.commuting-squares-of-identifications`](foundation.commuting-squares-of-identifications.md)                   |
+| Dependent identifications (foundation)            | [`foundation.dependent-identifications`](foundation.dependent-identifications.md)                                         |
+| Dependent identifications (foundation-core)       | [`foundation-core.dependent-identifications`](foundation-core.dependent-identifications.md)                               |
+| The fundamental theorem of identity types         | [`foundation.fundamental-theorem-of-identity-types`](foundation.fundamental-theorem-of-identity-types.md)                 |
+| Hexagons of identifications                       | [`foundation.hexagons-of-identifications`](foundation.hexagons-of-identifications.md)                                     |
+| Identity systems                                  | [`foundation.identity-systems`](foundation.identity-systems.md)                                                           |
+| The identity type (foundation)                    | [`foundation.identity-types`](foundation.identity-types.md)                                                               |
+| The identity type (foundation-core)               | [`foundation-core.identity-types`](foundation-core.identity-types.md)                                                     |
+| Large identity types                              | [`foundation.large-identity-types`](foundation.large-identity-types.md)                                                   |
+| Path algebra                                      | [`foundation.path-algebra`](foundation.path-algebra.md)                                                                   |
+| Symmetric identity types                          | [`foundation.symmetric-identity-types`](foundation.symmetric-identity-types.md)                                           |
+| Torsorial type families                           | [`foundation.torsorial-type-families`](foundation.torsorial-type-families.md)                                             |
+| Transport along identifications (foundation)      | [`foundation.transport-along-identifications`](foundation.transport-along-identifications.md)                             |
+| Transport along identifications (foundation-core) | [`foundation-core.transport-along-identifications`](foundation-core.transport-along-identifications.md)                   |
+| The universal property of identity types          | [`foundation.universal-property-identity-types`](foundation.universal-property-identity-types.md)                         |
 
 ## Definition
 
diff --git a/src/foundation/identity-types.lagda.md b/src/foundation/identity-types.lagda.md
index fd956e12c8..57e77a4bb7 100644
--- a/src/foundation/identity-types.lagda.md
+++ b/src/foundation/identity-types.lagda.md
@@ -37,27 +37,27 @@ is the least reflexive relation.
 The following table lists files that are about identity types and operations on
 identifications in arbitrary types.
 
-| Concept                                          | File                                                                                                                      |
-| ------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------- |
-| Action on identifications of binary functions    | [`foundation.action-on-identifications-binary-functions`](foundation.action-on-identifications-binary-functions.md)       |
-| Action on identifications of dependent functions | [`foundation.action-on-identifications-dependent-functions`](foundation.action-on-identifications-dependent-functions.md) |
-| Action on identifications of functions           | [`foundation.action-on-identifications-functions`](foundation.action-on-identifications-functions.md)                     |
-| Binary transport                                 | [`foundation.binary-transport`](foundation.binary-transport.md)                                                           |
-| Commuting squares of identifications             | [`foundation.commuting-squares-of-identifications`](foundation.commuting-squares-of-identifications.md)                   |
-| Dependent identifications (foundation)           | [`foundation.dependent-identifications`](foundation.dependent-identifications.md)                                         |
-| Dependent identifications (foundation-core)      | [`foundation-core.dependent-identifications`](foundation-core.dependent-identifications.md)                               |
-| The fundamental theorem of identity types        | [`foundation.fundamental-theorem-of-identity-types`](foundation.fundamental-theorem-of-identity-types.md)                 |
-| Hexagons of identifications                      | [`foundation.hexagons-of-identifications`](foundation.hexagons-of-identifications.md)                                     |
-| Identity systems                                 | [`foundation.identity-systems`](foundation.identity-systems.md)                                                           |
-| The identity type (foundation)                   | [`foundation.identity-types`](foundation.identity-types.md)                                                               |
-| The identity type (foundation-core)              | [`foundation-core.identity-types`](foundation-core.identity-types.md)                                                     |
-| Large identity types                             | [`foundation.large-identity-types`](foundation.large-identity-types.md)                                                   |
-| Path algebra                                     | [`foundation.path-algebra`](foundation.path-algebra.md)                                                                   |
-| Symmetric identity types                         | [`foundation.symmetric-identity-types`](foundation.symmetric-identity-types.md)                                           |
-| Torsorial type families                          | [`foundation.torsorial-type-families`](foundation.torsorial-type-families.md)                                             |
-| Transport (foundation)                           | [`foundation.transport-along-identifications`](foundation.transport-along-identifications.md)                             |
-| Transport (foundation-core)                      | [`foundation-core.transport-along-identifications`](foundation-core.transport-along-identifications.md)                   |
-| The universal property of identity types         | [`foundation.universal-property-identity-types`](foundation.universal-property-identity-types.md)                         |
+| Concept                                           | File                                                                                                                      |
+| ------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------- |
+| Action on identifications of binary functions     | [`foundation.action-on-identifications-binary-functions`](foundation.action-on-identifications-binary-functions.md)       |
+| Action on identifications of dependent functions  | [`foundation.action-on-identifications-dependent-functions`](foundation.action-on-identifications-dependent-functions.md) |
+| Action on identifications of functions            | [`foundation.action-on-identifications-functions`](foundation.action-on-identifications-functions.md)                     |
+| Binary transport                                  | [`foundation.binary-transport`](foundation.binary-transport.md)                                                           |
+| Commuting squares of identifications              | [`foundation.commuting-squares-of-identifications`](foundation.commuting-squares-of-identifications.md)                   |
+| Dependent identifications (foundation)            | [`foundation.dependent-identifications`](foundation.dependent-identifications.md)                                         |
+| Dependent identifications (foundation-core)       | [`foundation-core.dependent-identifications`](foundation-core.dependent-identifications.md)                               |
+| The fundamental theorem of identity types         | [`foundation.fundamental-theorem-of-identity-types`](foundation.fundamental-theorem-of-identity-types.md)                 |
+| Hexagons of identifications                       | [`foundation.hexagons-of-identifications`](foundation.hexagons-of-identifications.md)                                     |
+| Identity systems                                  | [`foundation.identity-systems`](foundation.identity-systems.md)                                                           |
+| The identity type (foundation)                    | [`foundation.identity-types`](foundation.identity-types.md)                                                               |
+| The identity type (foundation-core)               | [`foundation-core.identity-types`](foundation-core.identity-types.md)                                                     |
+| Large identity types                              | [`foundation.large-identity-types`](foundation.large-identity-types.md)                                                   |
+| Path algebra                                      | [`foundation.path-algebra`](foundation.path-algebra.md)                                                                   |
+| Symmetric identity types                          | [`foundation.symmetric-identity-types`](foundation.symmetric-identity-types.md)                                           |
+| Torsorial type families                           | [`foundation.torsorial-type-families`](foundation.torsorial-type-families.md)                                             |
+| Transport along identifications (foundation)      | [`foundation.transport-along-identifications`](foundation.transport-along-identifications.md)                             |
+| Transport along identifications (foundation-core) | [`foundation-core.transport-along-identifications`](foundation-core.transport-along-identifications.md)                   |
+| The universal property of identity types          | [`foundation.universal-property-identity-types`](foundation.universal-property-identity-types.md)                         |
 
 ## Properties
 

From 812965b9645a2a6be6fe06081bd013af687d7925 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 16:43:40 +0200
Subject: [PATCH 10/37] more lemmas for `tr-equiv`

---
 .../transport-along-identifications.lagda.md  |  3 +-
 .../transport-along-equivalences.lagda.md     | 56 +++++++++++++++----
 2 files changed, 46 insertions(+), 13 deletions(-)

diff --git a/src/foundation-core/transport-along-identifications.lagda.md b/src/foundation-core/transport-along-identifications.lagda.md
index 176a78d957..774aeec0c1 100644
--- a/src/foundation-core/transport-along-identifications.lagda.md
+++ b/src/foundation-core/transport-along-identifications.lagda.md
@@ -9,6 +9,7 @@ module foundation-core.transport-along-identifications where
 ```agda
 open import foundation.universe-levels
 
+open import foundation-core.function-types
 open import foundation-core.identity-types
 ```
 
@@ -32,7 +33,7 @@ recorded in
 ```agda
 tr :
   {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {x y : A} (p : x ＝ y) → B x → B y
-tr B refl b = b
+tr B refl = id
 ```
 
 ## Properties
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index fd45f6d1d1..b278f408e1 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -11,6 +11,7 @@ open import foundation.action-on-equivalences-functions
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
+open import foundation.equivalence-extensionality
 open import foundation.equivalences
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.transport-along-identifications
@@ -48,7 +49,7 @@ is-equiv-tr-equiv f {X} {Y} e = is-equiv-tr f (eq-equiv X Y e)
 
 equiv-tr-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
-  X ≃ Y → (f X) ≃ (f Y)
+  X ≃ Y → f X ≃ f Y
 pr1 (equiv-tr-equiv f e) = tr-equiv f e
 pr2 (equiv-tr-equiv f e) = is-equiv-tr-equiv f e
 ```
@@ -60,7 +61,7 @@ equiv-tr-equiv-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X : UU l1} →
   equiv-tr-equiv f id-equiv ＝ id-equiv
 equiv-tr-equiv-id-equiv f {X} =
-  ap (equiv-tr f) (compute-eq-equiv-id-equiv X) ∙ equiv-tr-refl f
+  (ap (equiv-tr f) (compute-eq-equiv-id-equiv X)) ∙ (equiv-tr-refl f)
 ```
 
 ### Transport along equivalences preserves composition of equivalences
@@ -75,6 +76,30 @@ tr-equiv-equiv-comp f {X} {Y} {Z} e e' x =
   ( tr-concat (eq-equiv X Y e) (eq-equiv Y Z e') x)
 ```
 
+### Transporting along an equivalence and its inverse is just the identity
+
+```agda
+is-section-tr-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2)
+  {X Y : UU l1} (e : X ≃ Y) →
+  (tr-equiv f (inv-equiv e) ∘ tr-equiv f e) ~ id
+is-section-tr-equiv f {X} {Y} e x =
+  ( ap
+    ( λ p → tr f p (tr-equiv f e x))
+    ( inv (commutativity-inv-eq-equiv X Y e))) ∙
+  ( is-section-tr f (eq-equiv X Y e) x)
+
+is-retraction-tr-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2)
+  {X Y : UU l1} (e : X ≃ Y) →
+  (tr-equiv f e ∘ tr-equiv f (inv-equiv e)) ~ id
+is-retraction-tr-equiv f {X} {Y} e x =
+  ( ap
+    ( tr-equiv f e ∘ (λ p → tr f p x))
+    ( inv (commutativity-inv-eq-equiv X Y e))) ∙
+  ( is-retraction-tr f (eq-equiv X Y e) x)
+```
+
 ### Transposing transport along the inverse of an equivalence
 
 ```agda
@@ -82,21 +107,15 @@ eq-transpose-tr-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2)
   {X Y : UU l1} (e : X ≃ Y) {u : f X} {v : f Y} →
   v ＝ tr-equiv f e u → tr-equiv f (inv-equiv e) v ＝ u
-eq-transpose-tr-equiv f {X} {Y} e {u} {v} p =
-  ( ap (tr-equiv f (inv-equiv e)) p) ∙
-  ( ( ap
-      ( λ q → tr f q (tr-equiv f e u))
-      ( inv (commutativity-inv-eq-equiv X Y e))) ∙
-    ( is-section-tr f (eq-equiv X Y e) u))
+eq-transpose-tr-equiv f e {u} p =
+  (ap (tr-equiv f (inv-equiv e)) p) ∙ (is-section-tr-equiv f e u)
 
 eq-transpose-tr-equiv' :
   {l1 l2 : Level} (f : UU l1 → UU l2)
   {X Y : UU l1} (e : X ≃ Y) {u : f X} {v : f Y} →
   tr-equiv f e u ＝ v → u ＝ tr-equiv f (inv-equiv e) v
-eq-transpose-tr-equiv' f {X} {Y} e {u} {v} p =
-  ( inv (is-section-tr f (eq-equiv X Y e) u)) ∙
-  ( ( ap (tr f (inv (eq-equiv X Y e))) p) ∙
-    ( ap (λ q → tr f q v) (commutativity-inv-eq-equiv X Y e)))
+eq-transpose-tr-equiv' f e {u} p =
+  (inv (is-section-tr-equiv f e u)) ∙ (ap (tr-equiv f (inv-equiv e)) p)
 ```
 
 ### Substitution law for transport along equivalences
@@ -109,3 +128,16 @@ tr-equiv-subst g f {X} {Y} e X' =
   ( ap (λ p → tr g p X') (is-retraction-eq-equiv (ap-eq-equiv f e))) ∙
   ( tr-subst g f (eq-equiv X Y e))
 ```
+
+### Transporting along the action on equivalences of a function
+
+```agda
+compute-tr-equiv-ap-equiv :
+  {l1 l2 l3 l4 : Level} {B : UU l1 → UU l2} {D : UU l3 → UU l4}
+  (f : UU l1 → UU l3) (g : (X : UU l1) → B X → D (f X))
+  {X Y : UU l1} (e : X ≃ Y) (X' : B X) →
+  tr-equiv D (ap-equiv f e) (g X X') ＝ g Y (tr-equiv B e X')
+compute-tr-equiv-ap-equiv {D = D} f g {X} {Y} e X' =
+  ( ap (λ p → tr D p (g X X')) (is-retraction-eq-equiv (ap-eq-equiv f e))) ∙
+  ( tr-ap f g (eq-equiv X Y e) X')
+```

From 89545cd7b91bd2d90d098f86518fc20b83df0750 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 17:01:24 +0200
Subject: [PATCH 11/37] The transport equivalence in the universe and action on
 equivalences coincide

---
 .../transport-along-identifications.lagda.md  |  2 +-
 .../transport-along-equivalences.lagda.md     | 23 +++++++++++++++++++
 2 files changed, 24 insertions(+), 1 deletion(-)

diff --git a/src/foundation-core/transport-along-identifications.lagda.md b/src/foundation-core/transport-along-identifications.lagda.md
index 774aeec0c1..a445055509 100644
--- a/src/foundation-core/transport-along-identifications.lagda.md
+++ b/src/foundation-core/transport-along-identifications.lagda.md
@@ -33,7 +33,7 @@ recorded in
 ```agda
 tr :
   {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {x y : A} (p : x ＝ y) → B x → B y
-tr B refl = id
+tr B refl b = b
 ```
 
 ## Properties
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index b278f408e1..11397dafdb 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -12,7 +12,9 @@ open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
 open import foundation.equivalence-extensionality
+open import foundation.equivalence-induction
 open import foundation.equivalences
+open import foundation.function-extensionality
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.transport-along-identifications
 open import foundation.univalence
@@ -141,3 +143,24 @@ compute-tr-equiv-ap-equiv {D = D} f g {X} {Y} e X' =
   ( ap (λ p → tr D p (g X X')) (is-retraction-eq-equiv (ap-eq-equiv f e))) ∙
   ( tr-ap f g (eq-equiv X Y e) X')
 ```
+
+### The transport equivalence in the universe and action on equivalences coincide
+
+```agda
+htpy-tr-equiv-ap-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
+  (e : X ≃ Y) → tr-equiv f e ~ map-equiv (ap-equiv f e)
+htpy-tr-equiv-ap-equiv f {X} {Y} e X' =
+  ind-equiv
+    ( X)
+    ( λ Y e' → tr-equiv f e' X' ＝ map-equiv (ap-equiv f e') X')
+    ( ( ap (λ p → tr f p X') (compute-eq-equiv-id-equiv X)) ∙
+      ( htpy-eq (ap pr1 (inv (ap-equiv-id-equiv f X))) X'))
+    ( e)
+
+eq-tr-equiv-ap-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
+  equiv-tr-equiv f {X} {Y} ~ ap-equiv f {X} {Y}
+eq-tr-equiv-ap-equiv f e =
+  eq-htpy-equiv (htpy-tr-equiv-ap-equiv f e)
+```

From 657201ac80f1ee0cc54fcf42461c5ceae184796f Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 17:03:49 +0200
Subject: [PATCH 12/37] delete `modal-type-theory`

---
 src/modal-type-theory.lagda.md | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
 delete mode 100644 src/modal-type-theory.lagda.md

diff --git a/src/modal-type-theory.lagda.md b/src/modal-type-theory.lagda.md
deleted file mode 100644
index e69de29bb2..0000000000

From e8faca9242f699009485c609a3e30bf09ad74d42 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 19:28:56 +0200
Subject: [PATCH 13/37] more elementary proof `eq-equiv-tr-equiv-ap-equiv`

---
 .../transport-along-identifications.lagda.md  |  1 -
 .../action-on-equivalences-functions.lagda.md |  8 -----
 .../transport-along-equivalences.lagda.md     | 32 +++++++------------
 .../function-classes.lagda.md                 |  2 +-
 4 files changed, 12 insertions(+), 31 deletions(-)

diff --git a/src/foundation-core/transport-along-identifications.lagda.md b/src/foundation-core/transport-along-identifications.lagda.md
index a445055509..176a78d957 100644
--- a/src/foundation-core/transport-along-identifications.lagda.md
+++ b/src/foundation-core/transport-along-identifications.lagda.md
@@ -9,7 +9,6 @@ module foundation-core.transport-along-identifications where
 ```agda
 open import foundation.universe-levels
 
-open import foundation-core.function-types
 open import foundation-core.identity-types
 ```
 
diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index 75740127a2..fe29c7e2cb 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -8,22 +8,14 @@ module foundation.action-on-equivalences-functions where
 
 ```agda
 open import foundation.action-on-identifications-functions
-open import foundation.dependent-pair-types
-open import foundation.equality-dependent-function-types
 open import foundation.equivalences
-open import foundation.function-extensionality
-open import foundation.fundamental-theorem-of-identity-types
-open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
 open import foundation-core.constant-maps
-open import foundation-core.contractible-types
 open import foundation-core.function-types
-open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.injective-maps
 ```
 
 </details>
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 11397dafdb..59c4567998 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -10,22 +10,16 @@ module foundation.transport-along-equivalences where
 open import foundation.action-on-equivalences-functions
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
-open import foundation.equality-dependent-function-types
-open import foundation.equivalence-extensionality
 open import foundation.equivalence-induction
 open import foundation.equivalences
 open import foundation.function-extensionality
-open import foundation.fundamental-theorem-of-identity-types
 open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
-open import foundation-core.contractible-types
 open import foundation-core.function-types
-open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.injective-maps
 ```
 
 </details>
@@ -144,23 +138,19 @@ compute-tr-equiv-ap-equiv {D = D} f g {X} {Y} e X' =
   ( tr-ap f g (eq-equiv X Y e) X')
 ```
 
-### The transport equivalence in the universe and action on equivalences coincide
+### Transport along and the action on equivalences in the universe coincide
 
 ```agda
-htpy-tr-equiv-ap-equiv :
+eq-equiv-tr-equiv-ap-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
-  (e : X ≃ Y) → tr-equiv f e ~ map-equiv (ap-equiv f e)
-htpy-tr-equiv-ap-equiv f {X} {Y} e X' =
-  ind-equiv
-    ( X)
-    ( λ Y e' → tr-equiv f e' X' ＝ map-equiv (ap-equiv f e') X')
-    ( ( ap (λ p → tr f p X') (compute-eq-equiv-id-equiv X)) ∙
-      ( htpy-eq (ap pr1 (inv (ap-equiv-id-equiv f X))) X'))
-    ( e)
-
-eq-tr-equiv-ap-equiv :
+  (e : X ≃ Y) → equiv-tr-equiv f e ＝ ap-equiv f e
+eq-equiv-tr-equiv-ap-equiv f {X} =
+    ind-equiv X
+      ( λ Y e → equiv-tr-equiv f e ＝ ap-equiv f e)
+      ( equiv-tr-equiv-id-equiv f ∙ inv (ap-equiv-id-equiv f X))
+
+eq-tr-equiv-map-ap-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
-  equiv-tr-equiv f {X} {Y} ~ ap-equiv f {X} {Y}
-eq-tr-equiv-ap-equiv f e =
-  eq-htpy-equiv (htpy-tr-equiv-ap-equiv f e)
+  (e : X ≃ Y) → tr-equiv f e ＝ map-equiv (ap-equiv f e)
+eq-tr-equiv-map-ap-equiv f e = ap pr1 (eq-equiv-tr-equiv-ap-equiv f e)
 ```
diff --git a/src/orthogonal-factorization-systems/function-classes.lagda.md b/src/orthogonal-factorization-systems/function-classes.lagda.md
index 56e789dbf9..9c30c83a67 100644
--- a/src/orthogonal-factorization-systems/function-classes.lagda.md
+++ b/src/orthogonal-factorization-systems/function-classes.lagda.md
@@ -14,7 +14,7 @@ open import foundation.function-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.pullback-squares
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 ```

From 1268f316ca09c92032986205e0140c02a0c9cbb6 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 19:46:22 +0200
Subject: [PATCH 14/37] add some prose

---
 .../action-on-equivalences-functions.lagda.md  | 18 ++++++++++++++++--
 .../transport-along-equivalences.lagda.md      | 16 +++++++++++++++-
 2 files changed, 31 insertions(+), 3 deletions(-)

diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index fe29c7e2cb..c254e6c8bb 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -20,6 +20,20 @@ open import foundation-core.identity-types
 
 </details>
 
+## Idea
+
+Applying the
+[action on identifications](foundation.action-on-identifications-functions.md)
+to [identifications](foundation-core.identity-types.md) arising from the
+[univalence axiom](foundation.univalence.md) gives us the **action on
+equivalences**.
+
+Alternatively, one can apply
+[transport along identifications](foundation-core.transport-along-identifications.md)
+to get
+[transport along equivalences](foundation.transport-along-equivalences.md), but
+luckily, these two notions coincide.
+
 ## Definition
 
 ```agda
@@ -67,12 +81,12 @@ ap-equiv-comp g f {X} {Y} e =
 ```agda
 ap-eq-equiv-id-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) (A : UU l1) →
-  (ap-eq-equiv f (id-equiv {A = A})) ＝ refl
+  (ap-eq-equiv f id-equiv) ＝ refl
 ap-eq-equiv-id-equiv f A = ap (ap f) (compute-eq-equiv-id-equiv A)
 
 ap-equiv-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) (A : UU l1) →
-  (ap-equiv f (id-equiv {A = A})) ＝ id-equiv {A = f A}
+  (ap-equiv f id-equiv) ＝ id-equiv
 ap-equiv-id-equiv f A = ap equiv-eq (ap-eq-equiv-id-equiv f A)
 ```
 
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 59c4567998..4830b2c93a 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -24,6 +24,20 @@ open import foundation-core.identity-types
 
 </details>
 
+Applying
+[transport along identifications](foundation-core.transport-along-identifications.md)
+to [identifications](foundation-core.identity-types.md) arising from the
+[univalence axiom](foundation.univalence.md) gives us **transport along
+equivalences**.
+
+Since transport defines [equivalences](foundation-core.equivalences.md) of
+[fibers](foundation-core.fibers-of-maps.md), this gives us an _action on
+equivalences_. Alternatively, we could apply the
+[action on identifications](foundation.action-on-identifications-functions.md)
+to get another
+[action on equivalences](foundation.action-on-equivalences-functions.md), but
+luckily, these two notions coincide.
+
 ## Definition
 
 ```agda
@@ -138,7 +152,7 @@ compute-tr-equiv-ap-equiv {D = D} f g {X} {Y} e X' =
   ( tr-ap f g (eq-equiv X Y e) X')
 ```
 
-### Transport along and the action on equivalences in the universe coincide
+### Transport along equivalences and the action on equivalences in the universe coincide
 
 ```agda
 eq-equiv-tr-equiv-ap-equiv :

From 05b61374cbdd2c85e1e7942ea4cf82ca04bd5e08 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 19:56:12 +0200
Subject: [PATCH 15/37] surface refactor `univalence-action-on-equivalences`

---
 ...rtier-delooping-sign-homomorphism.lagda.md | 20 ++++----
 .../delooping-sign-homomorphism.lagda.md      | 12 ++---
 ...mpson-delooping-sign-homomorphism.lagda.md | 20 ++++----
 src/foundation.lagda.md                       |  2 +-
 ...-along-equivalences-subuniverses.lagda.md} | 46 ++++++++++---------
 5 files changed, 51 insertions(+), 49 deletions(-)
 rename src/foundation/{univalence-action-on-equivalences.lagda.md => transport-along-equivalences-subuniverses.lagda.md} (74%)

diff --git a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
index 18ce157974..f8df967777 100644
--- a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
@@ -27,9 +27,9 @@ open import foundation.mere-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
+open import foundation.transport-along-equivalences-subuniverses
 open import foundation.transport-along-identifications
 open import foundation.unit-type
-open import foundation.univalence-action-on-equivalences
 open import foundation.universe-levels
 
 open import group-theory.concrete-groups
@@ -57,7 +57,7 @@ module _
   { l : Level}
   where
 
-  not-even-difference-univalent-action-equiv :
+  not-even-difference-ap-equiv-subuniverse :
     (n : ℕ) (Y : 2-Element-Decidable-Subtype l (raise-Fin l (n +ℕ 2))) →
     ¬ ( sim-Equivalence-Relation
       ( even-difference-orientation-Complete-Undirected-Graph
@@ -69,7 +69,7 @@ module _
         ( star)
         ( transposition Y))
       ( map-equiv
-        ( univalent-action-equiv
+        ( ap-equiv-subuniverse
           ( mere-equiv-Prop (Fin (n +ℕ 2)))
           ( orientation-Complete-Undirected-Graph (n +ℕ 2))
           ( raise l (Fin (n +ℕ 2)) ,
@@ -79,7 +79,7 @@ module _
           ( transposition Y))
         ( orientation-aut-count
           (n +ℕ 2 , compute-raise l (Fin (n +ℕ 2))) star (transposition Y))))
-  not-even-difference-univalent-action-equiv n =
+  not-even-difference-ap-equiv-subuniverse n =
     tr
       ( λ f →
         ( Y : 2-Element-Decidable-Subtype l
@@ -110,14 +110,14 @@ module _
           preserves-id-equiv-orientation-complete-undirected-graph-equiv
             ( n +ℕ 2)}
         { y =
-          ( univalent-action-equiv
+          ( ap-equiv-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( orientation-Complete-Undirected-Graph (n +ℕ 2))) ,
-          ( preserves-id-equiv-univalent-action-equiv
+          ( preserves-id-equiv-ap-equiv-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( orientation-Complete-Undirected-Graph (n +ℕ 2)))}
         ( eq-is-contr
-          ( is-contr-preserves-id-action-equiv
+          ( is-contr-preserves-id-ap-equiv-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( orientation-Complete-Undirected-Graph (n +ℕ 2))
             ( is-set-orientation-Complete-Undirected-Graph (n +ℕ 2)))))
@@ -135,7 +135,7 @@ module _
       ( equiv-fin-2-quotient-sign-equiv-Fin)
       ( λ n →
         orientation-aut-count (n +ℕ 2 , compute-raise l (Fin (n +ℕ 2))) (star))
-      ( not-even-difference-univalent-action-equiv)
+      ( not-even-difference-ap-equiv-subuniverse)
 
   eq-cartier-delooping-sign-homomorphism :
     (n : ℕ) →
@@ -176,7 +176,7 @@ module _
               orientation-aut-count
                 ( n +ℕ 2 , compute-raise l (Fin (n +ℕ 2)))
                 ( star))
-            ( not-even-difference-univalent-action-equiv)
+            ( not-even-difference-ap-equiv-subuniverse)
             ( n))
           ( sign-homomorphism
             ( n +ℕ 2)
@@ -194,7 +194,7 @@ module _
       ( equiv-fin-2-quotient-sign-equiv-Fin)
       ( λ n →
         orientation-aut-count (n +ℕ 2 , compute-raise l (Fin (n +ℕ 2))) (star))
-      ( not-even-difference-univalent-action-equiv)
+      ( not-even-difference-ap-equiv-subuniverse)
 ```
 
 ## References
diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index 8034a561a9..9ffb5d7e18 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -47,12 +47,12 @@ open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.reflecting-maps-equivalence-relations
 open import foundation.sets
+open import foundation.transport-along-equivalences-subuniverses
 open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.uniqueness-set-quotients
 open import foundation.unit-type
 open import foundation.univalence
-open import foundation.univalence-action-on-equivalences
 open import foundation.universal-property-set-quotients
 open import foundation.universe-levels
 
@@ -129,7 +129,7 @@ module _
           unit-trunc-Prop (compute-raise-Fin l1 (n +ℕ 2))))
       ( quotient-aut-succ-succ-Fin n (transposition Y))
       ( map-equiv
-        ( univalent-action-equiv
+        ( ap-equiv-subuniverse
           ( mere-equiv-Prop (Fin (n +ℕ 2)))
           ( D (n +ℕ 2))
           ( raise l1 (Fin (n +ℕ 2)) ,
@@ -165,13 +165,13 @@ module _
       (n : ℕ) (X X' : UU-Fin l1 n) →
       type-UU-Fin n X ≃ type-UU-Fin n X' → D n X ≃ D n X'
     invertible-action-D-equiv n =
-      univalent-action-equiv (mere-equiv-Prop (Fin n)) (D n)
+      ap-equiv-subuniverse (mere-equiv-Prop (Fin n)) (D n)
 
     preserves-id-equiv-invertible-action-D-equiv :
       (n : ℕ) (X : UU-Fin l1 n) →
       Id (invertible-action-D-equiv n X X id-equiv) id-equiv
     preserves-id-equiv-invertible-action-D-equiv n =
-      preserves-id-equiv-univalent-action-equiv (mere-equiv-Prop (Fin n)) (D n)
+      preserves-id-equiv-ap-equiv-subuniverse (mere-equiv-Prop (Fin n)) (D n)
 
     preserves-R-invertible-action-D-equiv :
       ( n : ℕ) →
@@ -184,7 +184,7 @@ module _
           ( map-equiv (invertible-action-D-equiv n X X' e) a)
           ( map-equiv (invertible-action-D-equiv n X X' e) a'))
     preserves-R-invertible-action-D-equiv n X X' =
-      Ind-univalent-action-equiv (mere-equiv-Prop (Fin n)) (D n) X
+      Ind-ap-equiv-subuniverse (mere-equiv-Prop (Fin n)) (D n) X
         ( λ Y f →
           ( a a' : D n X) →
           ( sim-Equivalence-Relation (R n X) a a' ↔
@@ -1488,7 +1488,7 @@ module _
             unit-trunc-Prop (compute-raise-Fin l1 (n +ℕ 2))))) →
     ¬ ( x ＝
         ( map-equiv
-          ( univalent-action-equiv
+          ( ap-equiv-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( type-UU-Fin 2 ∘ Q (n +ℕ 2))
             ( raise l1 (Fin (n +ℕ 2)) ,
diff --git a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
index 1166b556df..a3eccdb8db 100644
--- a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
@@ -41,9 +41,9 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
 open import foundation.sets
+open import foundation.transport-along-equivalences-subuniverses
 open import foundation.transport-along-identifications
 open import foundation.unit-type
-open import foundation.univalence-action-on-equivalences
 open import foundation.universe-levels
 
 open import group-theory.concrete-groups
@@ -673,7 +673,7 @@ module _
     Fin (n +ℕ 2) ≃ raise l (Fin (n +ℕ 2))
   sign-comp-aut-succ-succ-Fin n = _∘e compute-raise l (Fin (n +ℕ 2))
 
-  not-univalent-action-equiv-transposition :
+  not-ap-equiv-subuniverse-transposition :
     ( n : ℕ) →
     ( Y : 2-Element-Decidable-Subtype l
       ( raise-Fin l (n +ℕ 2))) →
@@ -683,7 +683,7 @@ module _
           unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2))))
       ( sign-comp-aut-succ-succ-Fin n (transposition Y))
       ( map-equiv
-        ( univalent-action-equiv
+        ( ap-equiv-subuniverse
           ( mere-equiv-Prop (Fin (n +ℕ 2)))
           ( λ X → Fin (n +ℕ 2) ≃ pr1 X)
           ( raise l (Fin (n +ℕ 2)) ,
@@ -692,7 +692,7 @@ module _
             unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2)))
           ( transposition Y))
         ( sign-comp-aut-succ-succ-Fin n (transposition Y))))
-  not-univalent-action-equiv-transposition n =
+  not-ap-equiv-subuniverse-transposition n =
     tr
       ( λ f →
         ( Y : 2-Element-Decidable-Subtype l
@@ -716,14 +716,14 @@ module _
           simpson-comp-equiv (n +ℕ 2) ,
           preserves-id-equiv-simpson-comp-equiv (n +ℕ 2)}
         { y =
-          ( univalent-action-equiv
+          ( ap-equiv-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X) ,
-            ( preserves-id-equiv-univalent-action-equiv
+            ( preserves-id-equiv-ap-equiv-subuniverse
               ( mere-equiv-Prop (Fin (n +ℕ 2)))
               ( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X)))}
         ( eq-is-contr
-          ( is-contr-preserves-id-action-equiv
+          ( is-contr-preserves-id-ap-equiv-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X)
             ( λ X →
@@ -743,7 +743,7 @@ module _
       ( λ n _ → is-decidable-sign-comp-Equivalence-Relation n)
       ( equiv-fin-2-quotient-sign-comp-equiv-Fin)
       ( sign-comp-aut-succ-succ-Fin)
-      ( not-univalent-action-equiv-transposition)
+      ( not-ap-equiv-subuniverse-transposition)
 
   eq-simpson-delooping-sign-homomorphism :
     (n : ℕ) →
@@ -779,7 +779,7 @@ module _
             ( λ n H → is-decidable-sign-comp-Equivalence-Relation n)
             ( equiv-fin-2-quotient-sign-comp-equiv-Fin)
             ( sign-comp-aut-succ-succ-Fin)
-            ( not-univalent-action-equiv-transposition)
+            ( not-ap-equiv-subuniverse-transposition)
             ( n))
           ( sign-homomorphism
             ( n +ℕ 2)
@@ -795,7 +795,7 @@ module _
       ( λ n _ → is-decidable-sign-comp-Equivalence-Relation n)
       ( equiv-fin-2-quotient-sign-comp-equiv-Fin)
       ( sign-comp-aut-succ-succ-Fin)
-      ( not-univalent-action-equiv-transposition)
+      ( not-ap-equiv-subuniverse-transposition)
 ```
 
 ## References
diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index 60e18c55fc..3bfffa2a30 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -255,6 +255,7 @@ open import foundation.symmetric-operations public
 open import foundation.tight-apartness-relations public
 open import foundation.torsorial-type-families public
 open import foundation.transport-along-equivalences public
+open import foundation.transport-along-equivalences-subuniverses public
 open import foundation.transport-along-identifications public
 open import foundation.trivial-relaxed-sigma-decompositions public
 open import foundation.trivial-sigma-decompositions public
@@ -285,7 +286,6 @@ open import foundation.uniqueness-truncation public
 open import foundation.unit-type public
 open import foundation.unital-binary-operations public
 open import foundation.univalence public
-open import foundation.univalence-action-on-equivalences public
 open import foundation.univalence-implies-function-extensionality public
 open import foundation.univalent-type-families public
 open import foundation.universal-property-booleans public
diff --git a/src/foundation/univalence-action-on-equivalences.lagda.md b/src/foundation/transport-along-equivalences-subuniverses.lagda.md
similarity index 74%
rename from src/foundation/univalence-action-on-equivalences.lagda.md
rename to src/foundation/transport-along-equivalences-subuniverses.lagda.md
index 6eaf2eafcb..99c1cc6322 100644
--- a/src/foundation/univalence-action-on-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences-subuniverses.lagda.md
@@ -1,7 +1,7 @@
-# Univalent action on equivalences
+# Transport along equivalences in subuniverses
 
 ```agda
-module foundation.univalence-action-on-equivalences where
+module foundation.transport-along-equivalences-subuniverses where
 ```
 
 <details><summary>Imports</summary>
@@ -29,8 +29,10 @@ open import foundation-core.subtypes
 
 ## Ideas
 
-Given a subuniverse `P`, any family of types `B` indexed by types of `P` has an
-action on equivalences obtained by using the univalence axiom.
+Given a [subuniverse](foundation.subuniverses.md) `P`, any family of types `B`
+indexed by types of `P` has an
+[action on equivalences](foundation.action-on-equivalences.md) obtained by using
+the [univalence axiom](foundation.univalence.md).
 
 ## Definition
 
@@ -40,18 +42,18 @@ module _
   ( P : subuniverse l1 l2) (B : type-subuniverse P → UU l3)
   where
 
-  univalent-action-equiv :
+  ap-equiv-subuniverse :
     (X Y : type-subuniverse P) → pr1 X ≃ pr1 Y → B X ≃ B Y
-  univalent-action-equiv X Y e = equiv-tr B (eq-equiv-subuniverse P e)
+  ap-equiv-subuniverse X Y e = equiv-tr B (eq-equiv-subuniverse P e)
 ```
 
 ## Properties
 
 ```agda
-  preserves-id-equiv-univalent-action-equiv :
+  preserves-id-equiv-ap-equiv-subuniverse :
     (X : type-subuniverse P) →
-    univalent-action-equiv X X id-equiv ＝ id-equiv
-  preserves-id-equiv-univalent-action-equiv X =
+    ap-equiv-subuniverse X X id-equiv ＝ id-equiv
+  preserves-id-equiv-ap-equiv-subuniverse X =
     ( ap (equiv-tr B)
       ( is-injective-map-equiv
         ( extensionality-subuniverse P X X)
@@ -60,38 +62,38 @@ module _
           ( id-equiv)))) ∙
       ( equiv-tr-refl B)
 
-  Ind-univalent-action-path :
+  Ind-path-ap-equiv-subuniverse :
     { l4 : Level}
     ( X : type-subuniverse P)
     ( F : (Y : type-subuniverse P) → B X ≃ B Y → UU l4) →
     F X id-equiv → ( Y : type-subuniverse P) (p : X ＝ Y) →
     F Y (equiv-tr B p)
-  Ind-univalent-action-path X F p .X refl =
+  Ind-path-ap-equiv-subuniverse X F p .X refl =
     tr (F X) (inv (equiv-tr-refl B)) p
 
-  Ind-univalent-action-equiv :
+  Ind-ap-equiv-subuniverse :
     { l4 : Level}
     ( X : type-subuniverse P)
     ( F : (Y : type-subuniverse P) → B X ≃ B Y → UU l4) →
     F X id-equiv → (Y : type-subuniverse P) (e : pr1 X ≃ pr1 Y) →
-    F Y (univalent-action-equiv X Y e)
-  Ind-univalent-action-equiv X F p Y e =
-    Ind-univalent-action-path X F p Y (eq-equiv-subuniverse P e)
+    F Y (ap-equiv-subuniverse X Y e)
+  Ind-ap-equiv-subuniverse X F p Y e =
+    Ind-path-ap-equiv-subuniverse X F p Y (eq-equiv-subuniverse P e)
 
-  is-contr-preserves-id-action-equiv :
+  is-contr-preserves-id-ap-equiv-subuniverse :
     ( (X : type-subuniverse P) → is-set (B X)) →
     is-contr
       ( Σ
         ( (X Y : type-subuniverse P) → pr1 X ≃ pr1 Y → B X ≃ B Y)
         ( λ D → (X : type-subuniverse P) → D X X id-equiv ＝ id-equiv))
-  pr1 (pr1 (is-contr-preserves-id-action-equiv H)) = univalent-action-equiv
-  pr2 (pr1 (is-contr-preserves-id-action-equiv H)) =
-    preserves-id-equiv-univalent-action-equiv
-  pr2 (is-contr-preserves-id-action-equiv H) (D , p) =
+  pr1 (pr1 (is-contr-preserves-id-ap-equiv-subuniverse H)) = ap-equiv-subuniverse
+  pr2 (pr1 (is-contr-preserves-id-ap-equiv-subuniverse H)) =
+    preserves-id-equiv-ap-equiv-subuniverse
+  pr2 (is-contr-preserves-id-ap-equiv-subuniverse H) (D , p) =
     eq-pair-Σ
       ( eq-htpy (λ X → eq-htpy (λ Y → eq-htpy (λ e →
-        lemma2 univalent-action-equiv D
-          (λ X → preserves-id-equiv-univalent-action-equiv X ∙ inv (p X))
+        lemma2 ap-equiv-subuniverse D
+          (λ X → preserves-id-equiv-ap-equiv-subuniverse X ∙ inv (p X))
           X Y e))))
       ( eq-is-prop
         ( is-prop-Π

From ccb54141ec0aecd5c2f92a10c002bcf2f630fe78 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 19:57:55 +0200
Subject: [PATCH 16/37] pre-commit

---
 .../transport-along-equivalences-subuniverses.lagda.md         | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/src/foundation/transport-along-equivalences-subuniverses.lagda.md b/src/foundation/transport-along-equivalences-subuniverses.lagda.md
index 99c1cc6322..6f4f0b381d 100644
--- a/src/foundation/transport-along-equivalences-subuniverses.lagda.md
+++ b/src/foundation/transport-along-equivalences-subuniverses.lagda.md
@@ -86,7 +86,8 @@ module _
       ( Σ
         ( (X Y : type-subuniverse P) → pr1 X ≃ pr1 Y → B X ≃ B Y)
         ( λ D → (X : type-subuniverse P) → D X X id-equiv ＝ id-equiv))
-  pr1 (pr1 (is-contr-preserves-id-ap-equiv-subuniverse H)) = ap-equiv-subuniverse
+  pr1 (pr1 (is-contr-preserves-id-ap-equiv-subuniverse H)) =
+    ap-equiv-subuniverse
   pr2 (pr1 (is-contr-preserves-id-ap-equiv-subuniverse H)) =
     preserves-id-equiv-ap-equiv-subuniverse
   pr2 (is-contr-preserves-id-ap-equiv-subuniverse H) (D , p) =

From ab9c34b414c3b48e5846dc3d0a4d0126b47c7c1a Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 20:03:02 +0200
Subject: [PATCH 17/37] Add `Idea` section header

---
 src/foundation/transport-along-equivalences.lagda.md | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 4830b2c93a..bd588957ac 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -24,6 +24,8 @@ open import foundation-core.identity-types
 
 </details>
 
+## Idea
+
 Applying
 [transport along identifications](foundation-core.transport-along-identifications.md)
 to [identifications](foundation-core.identity-types.md) arising from the

From d32d6781a00fa3b76b7ab7317715995dbfe51fc6 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 20:06:48 +0200
Subject: [PATCH 18/37] `tr-equiv-id-equiv`

---
 src/foundation/transport-along-equivalences.lagda.md | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index bd588957ac..cf54e899cf 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -69,6 +69,11 @@ pr2 (equiv-tr-equiv f e) = is-equiv-tr-equiv f e
 ### Transporting along `id-equiv` is the identity equivalence
 
 ```agda
+tr-equiv-id-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X : UU l1} →
+  tr-equiv f id-equiv ＝ id
+tr-equiv-id-equiv f {X} = ap (tr f) (compute-eq-equiv-id-equiv X)
+
 equiv-tr-equiv-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X : UU l1} →
   equiv-tr-equiv f id-equiv ＝ id-equiv

From 9758df5bf76ad85b65c6b90520b96b5d45527b63 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 20:11:13 +0200
Subject: [PATCH 19/37] `equiv-tr-equiv-equiv-comp`

---
 src/foundation/transport-along-equivalences.lagda.md | 8 ++++++++
 1 file changed, 8 insertions(+)

diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index cf54e899cf..24aecaca21 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -10,6 +10,7 @@ module foundation.transport-along-equivalences where
 open import foundation.action-on-equivalences-functions
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
+open import foundation.equivalence-extensionality
 open import foundation.equivalence-induction
 open import foundation.equivalences
 open import foundation.function-extensionality
@@ -91,6 +92,13 @@ tr-equiv-equiv-comp :
 tr-equiv-equiv-comp f {X} {Y} {Z} e e' x =
   ( ap (λ p → tr f p x) (inv (compute-eq-equiv-equiv-comp X Y Z e e'))) ∙
   ( tr-concat (eq-equiv X Y e) (eq-equiv Y Z e') x)
+
+equiv-tr-equiv-equiv-comp :
+  {l1 l2 : Level} (f : UU l1 → UU l2)
+  {X Y Z : UU l1} (e : X ≃ Y) (e' : Y ≃ Z) →
+  equiv-tr-equiv f (e' ∘e e) ＝ (equiv-tr-equiv f e' ∘e equiv-tr-equiv f e)
+equiv-tr-equiv-equiv-comp f {X} {Y} {Z} e e' =
+  eq-htpy-equiv (tr-equiv-equiv-comp f e e')
 ```
 
 ### Transporting along an equivalence and its inverse is just the identity

From 5aa3dc1baea902814fe5d60de523f8162fba59ed Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 20:21:39 +0200
Subject: [PATCH 20/37] `equiv-tr-equiv-subst`

---
 src/foundation/transport-along-equivalences.lagda.md | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 24aecaca21..e7b40cddc7 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -152,6 +152,11 @@ tr-equiv-subst :
 tr-equiv-subst g f {X} {Y} e X' =
   ( ap (λ p → tr g p X') (is-retraction-eq-equiv (ap-eq-equiv f e))) ∙
   ( tr-subst g f (eq-equiv X Y e))
+
+equiv-tr-equiv-subst :
+  {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2) {X Y : UU l1}
+  (e : X ≃ Y) → equiv-tr-equiv g (ap-equiv f e) ＝ equiv-tr-equiv (g ∘ f) e
+equiv-tr-equiv-subst g f e = eq-htpy-equiv (tr-equiv-subst g f e)
 ```
 
 ### Transporting along the action on equivalences of a function

From e531bb16aae518115b612c9924800550e9aadaa2 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 20:26:55 +0200
Subject: [PATCH 21/37] renamings `transport-along-equivalences`

---
 .../equality-dependent-pair-types.lagda.md    |  2 +-
 .../transport-along-equivalences.lagda.md     | 30 +++++++++----------
 .../transport-along-identifications.lagda.md  |  4 +--
 3 files changed, 18 insertions(+), 18 deletions(-)

diff --git a/src/foundation/equality-dependent-pair-types.lagda.md b/src/foundation/equality-dependent-pair-types.lagda.md
index e5aa286fc1..426a1c39da 100644
--- a/src/foundation/equality-dependent-pair-types.lagda.md
+++ b/src/foundation/equality-dependent-pair-types.lagda.md
@@ -100,7 +100,7 @@ module _
     {x y : X} (p : x ＝ y) →
     pair-eq-Σ (ap f p) ＝
     ( ( ap (pr1 ∘ f) p) ,
-      ( tr-subst B (pr1 ∘ f) p ∙ apd (pr2 ∘ f) p))
+      ( substitution-tr B (pr1 ∘ f) p ∙ apd (pr2 ∘ f) p))
   pair-eq-Σ-ap refl = refl
 
   pr1-pair-eq-Σ-ap :
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index e7b40cddc7..b54e0c4bcc 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -70,35 +70,35 @@ pr2 (equiv-tr-equiv f e) = is-equiv-tr-equiv f e
 ### Transporting along `id-equiv` is the identity equivalence
 
 ```agda
-tr-equiv-id-equiv :
+compute-tr-equiv-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X : UU l1} →
   tr-equiv f id-equiv ＝ id
-tr-equiv-id-equiv f {X} = ap (tr f) (compute-eq-equiv-id-equiv X)
+compute-tr-equiv-id-equiv f {X} = ap (tr f) (compute-eq-equiv-id-equiv X)
 
-equiv-tr-equiv-id-equiv :
+compute-equiv-tr-equiv-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X : UU l1} →
   equiv-tr-equiv f id-equiv ＝ id-equiv
-equiv-tr-equiv-id-equiv f {X} =
+compute-equiv-tr-equiv-id-equiv f {X} =
   (ap (equiv-tr f) (compute-eq-equiv-id-equiv X)) ∙ (equiv-tr-refl f)
 ```
 
 ### Transport along equivalences preserves composition of equivalences
 
 ```agda
-tr-equiv-equiv-comp :
+distributive-tr-equiv-equiv-comp :
   {l1 l2 : Level} (f : UU l1 → UU l2)
   {X Y Z : UU l1} (e : X ≃ Y) (e' : Y ≃ Z) →
   tr-equiv f (e' ∘e e) ~ (tr-equiv f e' ∘ tr-equiv f e)
-tr-equiv-equiv-comp f {X} {Y} {Z} e e' x =
+distributive-tr-equiv-equiv-comp f {X} {Y} {Z} e e' x =
   ( ap (λ p → tr f p x) (inv (compute-eq-equiv-equiv-comp X Y Z e e'))) ∙
   ( tr-concat (eq-equiv X Y e) (eq-equiv Y Z e') x)
 
-equiv-tr-equiv-equiv-comp :
+distributive-equiv-tr-equiv-equiv-comp :
   {l1 l2 : Level} (f : UU l1 → UU l2)
   {X Y Z : UU l1} (e : X ≃ Y) (e' : Y ≃ Z) →
   equiv-tr-equiv f (e' ∘e e) ＝ (equiv-tr-equiv f e' ∘e equiv-tr-equiv f e)
-equiv-tr-equiv-equiv-comp f {X} {Y} {Z} e e' =
-  eq-htpy-equiv (tr-equiv-equiv-comp f e e')
+distributive-equiv-tr-equiv-equiv-comp f {X} {Y} {Z} e e' =
+  eq-htpy-equiv (distributive-tr-equiv-equiv-comp f e e')
 ```
 
 ### Transporting along an equivalence and its inverse is just the identity
@@ -146,17 +146,17 @@ eq-transpose-tr-equiv' f e {u} p =
 ### Substitution law for transport along equivalences
 
 ```agda
-tr-equiv-subst :
+substitution-tr-equiv :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2) {X Y : UU l1}
   (e : X ≃ Y) → tr-equiv g (ap-equiv f e) ~ tr-equiv (g ∘ f) e
-tr-equiv-subst g f {X} {Y} e X' =
+substitution-tr-equiv g f {X} {Y} e X' =
   ( ap (λ p → tr g p X') (is-retraction-eq-equiv (ap-eq-equiv f e))) ∙
-  ( tr-subst g f (eq-equiv X Y e))
+  ( substitution-tr g f (eq-equiv X Y e))
 
-equiv-tr-equiv-subst :
+substitution-equiv-tr-equiv :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2) {X Y : UU l1}
   (e : X ≃ Y) → equiv-tr-equiv g (ap-equiv f e) ＝ equiv-tr-equiv (g ∘ f) e
-equiv-tr-equiv-subst g f e = eq-htpy-equiv (tr-equiv-subst g f e)
+substitution-equiv-tr-equiv g f e = eq-htpy-equiv (substitution-tr-equiv g f e)
 ```
 
 ### Transporting along the action on equivalences of a function
@@ -181,7 +181,7 @@ eq-equiv-tr-equiv-ap-equiv :
 eq-equiv-tr-equiv-ap-equiv f {X} =
     ind-equiv X
       ( λ Y e → equiv-tr-equiv f e ＝ ap-equiv f e)
-      ( equiv-tr-equiv-id-equiv f ∙ inv (ap-equiv-id-equiv f X))
+      ( compute-equiv-tr-equiv-id-equiv f ∙ inv (ap-equiv-id-equiv f X))
 
 eq-tr-equiv-map-ap-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
diff --git a/src/foundation/transport-along-identifications.lagda.md b/src/foundation/transport-along-identifications.lagda.md
index 9b917a57da..514e21aa3f 100644
--- a/src/foundation/transport-along-identifications.lagda.md
+++ b/src/foundation/transport-along-identifications.lagda.md
@@ -88,9 +88,9 @@ equiv-tr-refl B = refl
 ### Substitution law for transport
 
 ```agda
-tr-subst :
+substitution-tr :
   {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} (B : A → UU l3) (f : X → A)
   {x y : X} (p : x ＝ y) {x' : B (f x)} →
   tr B (ap f p) x' ＝ tr (B ∘ f) p x'
-tr-subst B f refl = refl
+substitution-tr B f refl = refl
 ```

From 45fc523486321a6b21a0e8cea1a841a77b9bd56f Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 20:30:37 +0200
Subject: [PATCH 22/37] action* on* equivalences in subuniverse

---
 .../cartier-delooping-sign-homomorphism.lagda.md              | 2 +-
 src/finite-group-theory/delooping-sign-homomorphism.lagda.md  | 2 +-
 .../simpson-delooping-sign-homomorphism.lagda.md              | 2 +-
 src/foundation.lagda.md                                       | 2 +-
 ....lagda.md => action-on-equivalences-subuniverses.lagda.md} | 4 ++--
 5 files changed, 6 insertions(+), 6 deletions(-)
 rename src/foundation/{transport-along-equivalences-subuniverses.lagda.md => action-on-equivalences-subuniverses.lagda.md} (97%)

diff --git a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
index f8df967777..c342c02201 100644
--- a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
@@ -17,6 +17,7 @@ open import finite-group-theory.finite-type-groups
 open import finite-group-theory.sign-homomorphism
 open import finite-group-theory.transpositions
 
+open import foundation.action-on-equivalences-subuniverses
 open import foundation.action-on-identifications-functions
 open import foundation.contractible-types
 open import foundation.dependent-pair-types
@@ -27,7 +28,6 @@ open import foundation.mere-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
-open import foundation.transport-along-equivalences-subuniverses
 open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index 9ffb5d7e18..c00d5d7b7f 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -18,6 +18,7 @@ open import finite-group-theory.permutations
 open import finite-group-theory.sign-homomorphism
 open import finite-group-theory.transpositions
 
+open import foundation.action-on-equivalences-subuniverses
 open import foundation.action-on-identifications-functions
 open import foundation.commuting-squares-of-maps
 open import foundation.contractible-types
@@ -47,7 +48,6 @@ open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.reflecting-maps-equivalence-relations
 open import foundation.sets
-open import foundation.transport-along-equivalences-subuniverses
 open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.uniqueness-set-quotients
diff --git a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
index a3eccdb8db..64358bec43 100644
--- a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
@@ -21,6 +21,7 @@ open import finite-group-theory.permutations
 open import finite-group-theory.sign-homomorphism
 open import finite-group-theory.transpositions
 
+open import foundation.action-on-equivalences-subuniverses
 open import foundation.action-on-identifications-functions
 open import foundation.contractible-types
 open import foundation.coproduct-types
@@ -41,7 +42,6 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport-along-equivalences-subuniverses
 open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index 3bfffa2a30..84e6c7532b 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -11,6 +11,7 @@ open import foundation.0-maps public
 open import foundation.1-types public
 open import foundation.2-types public
 open import foundation.action-on-equivalences-functions public
+open import foundation.action-on-equivalences-subuniverses public
 open import foundation.action-on-identifications-binary-functions public
 open import foundation.action-on-identifications-dependent-functions public
 open import foundation.action-on-identifications-functions public
@@ -255,7 +256,6 @@ open import foundation.symmetric-operations public
 open import foundation.tight-apartness-relations public
 open import foundation.torsorial-type-families public
 open import foundation.transport-along-equivalences public
-open import foundation.transport-along-equivalences-subuniverses public
 open import foundation.transport-along-identifications public
 open import foundation.trivial-relaxed-sigma-decompositions public
 open import foundation.trivial-sigma-decompositions public
diff --git a/src/foundation/transport-along-equivalences-subuniverses.lagda.md b/src/foundation/action-on-equivalences-subuniverses.lagda.md
similarity index 97%
rename from src/foundation/transport-along-equivalences-subuniverses.lagda.md
rename to src/foundation/action-on-equivalences-subuniverses.lagda.md
index 6f4f0b381d..588b6c11be 100644
--- a/src/foundation/transport-along-equivalences-subuniverses.lagda.md
+++ b/src/foundation/action-on-equivalences-subuniverses.lagda.md
@@ -1,7 +1,7 @@
-# Transport along equivalences in subuniverses
+# Action on equivalences in subuniverses
 
 ```agda
-module foundation.transport-along-equivalences-subuniverses where
+module foundation.action-on-equivalences-subuniverses where
 ```
 
 <details><summary>Imports</summary>

From 1fa91d24a758729f93c9cfd73b2c2b5377cefea5 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 20:42:47 +0200
Subject: [PATCH 23/37] renamings `action-on-equivalences-functions`

---
 .../action-on-equivalences-functions.lagda.md | 50 +++++++++----------
 .../transport-along-equivalences.lagda.md     |  2 +-
 2 files changed, 26 insertions(+), 26 deletions(-)

diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index c254e6c8bb..3b78c6adbb 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -53,91 +53,91 @@ ap-equiv f = equiv-eq ∘ ap-eq-equiv f
 ### The identity function acts trivially on equivalences
 
 ```agda
-ap-equiv-id :
+compute-ap-equiv-id :
   {l : Level} {X Y : UU l} (e : X ≃ Y) → (ap-equiv id e) ＝ e
-ap-equiv-id {l} {X} {Y} e =
+compute-ap-equiv-id {l} {X} {Y} e =
   (ap equiv-eq (ap-id (eq-equiv X Y e))) ∙ (is-section-eq-equiv e)
 ```
 
 ### The action on equivalences of a composite function is the composite of the actions
 
 ```agda
-ap-eq-equiv-comp :
+distributive-ap-eq-equiv-comp :
   {l1 l2 l3 : Level} {C : UU l3} (g : UU l2 → C) (f : UU l1 → UU l2)
   {X Y : UU l1} → ap-eq-equiv (g ∘ f) ~ (ap-eq-equiv g ∘ ap-equiv f)
-ap-eq-equiv-comp g f {X} {Y} e =
+distributive-ap-eq-equiv-comp g f {X} {Y} e =
   ( ap-comp g f (eq-equiv X Y e)) ∙
   ( ap (ap g) (inv (is-retraction-eq-equiv (ap-eq-equiv f e))))
 
-ap-equiv-comp :
+distributive-ap-equiv-comp :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2)
   {X Y : UU l1} → ap-equiv (g ∘ f) ~ (ap-equiv g ∘ ap-equiv f)
-ap-equiv-comp g f {X} {Y} e =
-  ap equiv-eq (ap-eq-equiv-comp g f {X} {Y} e)
+distributive-ap-equiv-comp g f {X} {Y} e =
+  ap equiv-eq (distributive-ap-eq-equiv-comp g f {X} {Y} e)
 ```
 
 ### The action on equivalences of any map preserves `id-equiv`
 
 ```agda
-ap-eq-equiv-id-equiv :
+compute-ap-eq-equiv-id-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) (A : UU l1) →
   (ap-eq-equiv f id-equiv) ＝ refl
-ap-eq-equiv-id-equiv f A = ap (ap f) (compute-eq-equiv-id-equiv A)
+compute-ap-eq-equiv-id-equiv f A = ap (ap f) (compute-eq-equiv-id-equiv A)
 
-ap-equiv-id-equiv :
+compute-ap-equiv-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) (A : UU l1) →
   (ap-equiv f id-equiv) ＝ id-equiv
-ap-equiv-id-equiv f A = ap equiv-eq (ap-eq-equiv-id-equiv f A)
+compute-ap-equiv-id-equiv f A = ap equiv-eq (compute-ap-eq-equiv-id-equiv f A)
 ```
 
 ### The action on equivalences of any map preserves composition of equivalences
 
 ```agda
-ap-eq-equiv-equiv-comp :
+distributive-ap-eq-equiv-equiv-comp :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y Z : UU l1} →
   (e : X ≃ Y) (e' : Y ≃ Z) →
   ap-eq-equiv f (e' ∘e e) ＝ (ap-eq-equiv f e ∙ ap-eq-equiv f e')
-ap-eq-equiv-equiv-comp f {X} {Y} {Z} e e' =
+distributive-ap-eq-equiv-equiv-comp f {X} {Y} {Z} e e' =
     ( ap (ap f) (inv (compute-eq-equiv-equiv-comp X Y Z e e'))) ∙
     ( ap-concat f (eq-equiv X Y e) (eq-equiv Y Z e'))
 
-ap-equiv-equiv-comp :
+distributive-ap-equiv-equiv-comp :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y Z : UU l1} →
   (e : X ≃ Y) (e' : Y ≃ Z) →
   ap-equiv f (e' ∘e e) ＝ (ap-equiv f e' ∘e ap-equiv f e)
-ap-equiv-equiv-comp f {X} {Y} {Z} e e' =
-  ( ap equiv-eq (ap-eq-equiv-equiv-comp f e e')) ∙
+distributive-ap-equiv-equiv-comp f {X} {Y} {Z} e e' =
+  ( ap equiv-eq (distributive-ap-eq-equiv-equiv-comp f e e')) ∙
   ( inv (compute-equiv-eq-concat (ap-eq-equiv f e) (ap-eq-equiv f e')))
 ```
 
 ### The action on equivalences of any map preserves inverses
 
 ```agda
-ap-eq-equiv-inv :
+compute-ap-eq-equiv-inv-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y : UU l1}
   (e : X ≃ Y) → ap-eq-equiv f (inv-equiv e) ＝ inv (ap-eq-equiv f e)
-ap-eq-equiv-inv f {X} {Y} e =
+compute-ap-eq-equiv-inv-equiv f {X} {Y} e =
   ( ap (ap f) (inv (commutativity-inv-eq-equiv X Y e))) ∙
   ( ap-inv f (eq-equiv X Y e))
 
-ap-equiv-inv :
+compute-ap-equiv-inv-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1}
   (e : X ≃ Y) → ap-equiv f (inv-equiv e) ＝ inv-equiv (ap-equiv f e)
-ap-equiv-inv f {X} {Y} e =
-  ( ap equiv-eq (ap-eq-equiv-inv f e)) ∙
+compute-ap-equiv-inv-equiv f {X} {Y} e =
+  ( ap equiv-eq (compute-ap-eq-equiv-inv-equiv f e)) ∙
   ( inv (commutativity-inv-equiv-eq (f X) (f Y) (ap-eq-equiv f e)))
 ```
 
 ### The action on equivalences of a constant map is constant
 
 ```agda
-ap-eq-equiv-const :
+compute-ap-eq-equiv-const :
   {l1 l2 : Level} {B : UU l2} (b : B) {X Y : UU l1}
   (e : X ≃ Y) → (ap-eq-equiv (const (UU l1) B b) e) ＝ refl
-ap-eq-equiv-const b {X} {Y} e = ap-const b (eq-equiv X Y e)
+compute-ap-eq-equiv-const b {X} {Y} e = ap-const b (eq-equiv X Y e)
 
-ap-equiv-const :
+compute-ap-equiv-const :
   {l1 l2 : Level} (B : UU l2) {X Y : UU l1}
   (e : X ≃ Y) → (ap-equiv (const (UU l1) (UU l2) B) e) ＝ id-equiv
-ap-equiv-const B {X} {Y} e = ap equiv-eq (ap-eq-equiv-const B e)
+compute-ap-equiv-const B {X} {Y} e = ap equiv-eq (compute-ap-eq-equiv-const B e)
 ```
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index b54e0c4bcc..2436099603 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -181,7 +181,7 @@ eq-equiv-tr-equiv-ap-equiv :
 eq-equiv-tr-equiv-ap-equiv f {X} =
     ind-equiv X
       ( λ Y e → equiv-tr-equiv f e ＝ ap-equiv f e)
-      ( compute-equiv-tr-equiv-id-equiv f ∙ inv (ap-equiv-id-equiv f X))
+      ( compute-equiv-tr-equiv-id-equiv f ∙ inv (compute-ap-equiv-id-equiv f X))
 
 eq-tr-equiv-map-ap-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →

From cb4e696ae55f75bdb86e783e6ff992e95600141d Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 23 Aug 2023 22:01:22 +0200
Subject: [PATCH 24/37] possible renaming

---
 .../action-on-equivalences-functions.lagda.md |  57 ++++----
 .../transport-along-equivalences.lagda.md     | 125 +++++++++---------
 2 files changed, 93 insertions(+), 89 deletions(-)

diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index 3b78c6adbb..a28d652594 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -37,15 +37,20 @@ luckily, these two notions coincide.
 ## Definition
 
 ```agda
-ap-eq-equiv :
+eq-ap-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y : UU l1} →
   X ≃ Y → f X ＝ f Y
-ap-eq-equiv f {X} {Y} e = ap f (eq-equiv X Y e)
+eq-ap-equiv f {X} {Y} e = ap f (eq-equiv X Y e)
 
 ap-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
   X ≃ Y → f X ≃ f Y
-ap-equiv f = equiv-eq ∘ ap-eq-equiv f
+ap-equiv f = equiv-eq ∘ eq-ap-equiv f
+
+map-ap-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
+  X ≃ Y → f X → f Y
+map-ap-equiv f = map-equiv ∘ ap-equiv f
 ```
 
 ## Properties
@@ -62,42 +67,42 @@ compute-ap-equiv-id {l} {X} {Y} e =
 ### The action on equivalences of a composite function is the composite of the actions
 
 ```agda
-distributive-ap-eq-equiv-comp :
+distributive-eq-ap-equiv-comp :
   {l1 l2 l3 : Level} {C : UU l3} (g : UU l2 → C) (f : UU l1 → UU l2)
-  {X Y : UU l1} → ap-eq-equiv (g ∘ f) ~ (ap-eq-equiv g ∘ ap-equiv f)
-distributive-ap-eq-equiv-comp g f {X} {Y} e =
+  {X Y : UU l1} → eq-ap-equiv (g ∘ f) ~ (eq-ap-equiv g ∘ ap-equiv f)
+distributive-eq-ap-equiv-comp g f {X} {Y} e =
   ( ap-comp g f (eq-equiv X Y e)) ∙
-  ( ap (ap g) (inv (is-retraction-eq-equiv (ap-eq-equiv f e))))
+  ( ap (ap g) (inv (is-retraction-eq-equiv (eq-ap-equiv f e))))
 
 distributive-ap-equiv-comp :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2)
   {X Y : UU l1} → ap-equiv (g ∘ f) ~ (ap-equiv g ∘ ap-equiv f)
 distributive-ap-equiv-comp g f {X} {Y} e =
-  ap equiv-eq (distributive-ap-eq-equiv-comp g f {X} {Y} e)
+  ap equiv-eq (distributive-eq-ap-equiv-comp g f {X} {Y} e)
 ```
 
 ### The action on equivalences of any map preserves `id-equiv`
 
 ```agda
-compute-ap-eq-equiv-id-equiv :
+compute-eq-ap-equiv-id-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) (A : UU l1) →
-  (ap-eq-equiv f id-equiv) ＝ refl
-compute-ap-eq-equiv-id-equiv f A = ap (ap f) (compute-eq-equiv-id-equiv A)
+  (eq-ap-equiv f id-equiv) ＝ refl
+compute-eq-ap-equiv-id-equiv f A = ap (ap f) (compute-eq-equiv-id-equiv A)
 
 compute-ap-equiv-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) (A : UU l1) →
   (ap-equiv f id-equiv) ＝ id-equiv
-compute-ap-equiv-id-equiv f A = ap equiv-eq (compute-ap-eq-equiv-id-equiv f A)
+compute-ap-equiv-id-equiv f A = ap equiv-eq (compute-eq-ap-equiv-id-equiv f A)
 ```
 
 ### The action on equivalences of any map preserves composition of equivalences
 
 ```agda
-distributive-ap-eq-equiv-equiv-comp :
+distributive-eq-ap-equiv-equiv-comp :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y Z : UU l1} →
   (e : X ≃ Y) (e' : Y ≃ Z) →
-  ap-eq-equiv f (e' ∘e e) ＝ (ap-eq-equiv f e ∙ ap-eq-equiv f e')
-distributive-ap-eq-equiv-equiv-comp f {X} {Y} {Z} e e' =
+  eq-ap-equiv f (e' ∘e e) ＝ (eq-ap-equiv f e ∙ eq-ap-equiv f e')
+distributive-eq-ap-equiv-equiv-comp f {X} {Y} {Z} e e' =
     ( ap (ap f) (inv (compute-eq-equiv-equiv-comp X Y Z e e'))) ∙
     ( ap-concat f (eq-equiv X Y e) (eq-equiv Y Z e'))
 
@@ -106,17 +111,17 @@ distributive-ap-equiv-equiv-comp :
   (e : X ≃ Y) (e' : Y ≃ Z) →
   ap-equiv f (e' ∘e e) ＝ (ap-equiv f e' ∘e ap-equiv f e)
 distributive-ap-equiv-equiv-comp f {X} {Y} {Z} e e' =
-  ( ap equiv-eq (distributive-ap-eq-equiv-equiv-comp f e e')) ∙
-  ( inv (compute-equiv-eq-concat (ap-eq-equiv f e) (ap-eq-equiv f e')))
+  ( ap equiv-eq (distributive-eq-ap-equiv-equiv-comp f e e')) ∙
+  ( inv (compute-equiv-eq-concat (eq-ap-equiv f e) (eq-ap-equiv f e')))
 ```
 
 ### The action on equivalences of any map preserves inverses
 
 ```agda
-compute-ap-eq-equiv-inv-equiv :
+compute-eq-ap-equiv-inv-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y : UU l1}
-  (e : X ≃ Y) → ap-eq-equiv f (inv-equiv e) ＝ inv (ap-eq-equiv f e)
-compute-ap-eq-equiv-inv-equiv f {X} {Y} e =
+  (e : X ≃ Y) → eq-ap-equiv f (inv-equiv e) ＝ inv (eq-ap-equiv f e)
+compute-eq-ap-equiv-inv-equiv f {X} {Y} e =
   ( ap (ap f) (inv (commutativity-inv-eq-equiv X Y e))) ∙
   ( ap-inv f (eq-equiv X Y e))
 
@@ -124,20 +129,20 @@ compute-ap-equiv-inv-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1}
   (e : X ≃ Y) → ap-equiv f (inv-equiv e) ＝ inv-equiv (ap-equiv f e)
 compute-ap-equiv-inv-equiv f {X} {Y} e =
-  ( ap equiv-eq (compute-ap-eq-equiv-inv-equiv f e)) ∙
-  ( inv (commutativity-inv-equiv-eq (f X) (f Y) (ap-eq-equiv f e)))
+  ( ap equiv-eq (compute-eq-ap-equiv-inv-equiv f e)) ∙
+  ( inv (commutativity-inv-equiv-eq (f X) (f Y) (eq-ap-equiv f e)))
 ```
 
 ### The action on equivalences of a constant map is constant
 
 ```agda
-compute-ap-eq-equiv-const :
+compute-eq-ap-equiv-const :
   {l1 l2 : Level} {B : UU l2} (b : B) {X Y : UU l1}
-  (e : X ≃ Y) → (ap-eq-equiv (const (UU l1) B b) e) ＝ refl
-compute-ap-eq-equiv-const b {X} {Y} e = ap-const b (eq-equiv X Y e)
+  (e : X ≃ Y) → (eq-ap-equiv (const (UU l1) B b) e) ＝ refl
+compute-eq-ap-equiv-const b {X} {Y} e = ap-const b (eq-equiv X Y e)
 
 compute-ap-equiv-const :
   {l1 l2 : Level} (B : UU l2) {X Y : UU l1}
   (e : X ≃ Y) → (ap-equiv (const (UU l1) (UU l2) B) e) ＝ id-equiv
-compute-ap-equiv-const B {X} {Y} e = ap equiv-eq (compute-ap-eq-equiv-const B e)
+compute-ap-equiv-const B {X} {Y} e = ap equiv-eq (compute-eq-ap-equiv-const B e)
 ```
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 2436099603..c72dffff47 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -44,83 +44,82 @@ luckily, these two notions coincide.
 ## Definition
 
 ```agda
-tr-equiv :
+map-tr-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
   X ≃ Y → f X → f Y
-tr-equiv f {X} {Y} e = tr f (eq-equiv X Y e)
-```
-
-## Properties
-
-### Transport along an equivalence is an equivalence
+map-tr-equiv f {X} {Y} e = tr f (eq-equiv X Y e)
 
-```agda
-is-equiv-tr-equiv :
+is-equiv-map-tr-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1}
-  (e : X ≃ Y) → is-equiv (tr-equiv f e)
-is-equiv-tr-equiv f {X} {Y} e = is-equiv-tr f (eq-equiv X Y e)
+  (e : X ≃ Y) → is-equiv (map-tr-equiv f e)
+is-equiv-map-tr-equiv f {X} {Y} e = is-equiv-tr f (eq-equiv X Y e)
 
-equiv-tr-equiv :
+tr-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
   X ≃ Y → f X ≃ f Y
-pr1 (equiv-tr-equiv f e) = tr-equiv f e
-pr2 (equiv-tr-equiv f e) = is-equiv-tr-equiv f e
+pr1 (tr-equiv f e) = map-tr-equiv f e
+pr2 (tr-equiv f e) = is-equiv-map-tr-equiv f e
+
+eq-tr-equiv :
+  {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
+  X ≃ Y → f X ＝ f Y
+eq-tr-equiv f {X} {Y} = eq-equiv (f X) (f Y) ∘ tr-equiv f
 ```
 
 ### Transporting along `id-equiv` is the identity equivalence
 
 ```agda
-compute-tr-equiv-id-equiv :
+compute-map-tr-equiv-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X : UU l1} →
-  tr-equiv f id-equiv ＝ id
-compute-tr-equiv-id-equiv f {X} = ap (tr f) (compute-eq-equiv-id-equiv X)
+  map-tr-equiv f id-equiv ＝ id
+compute-map-tr-equiv-id-equiv f {X} = ap (tr f) (compute-eq-equiv-id-equiv X)
 
-compute-equiv-tr-equiv-id-equiv :
+compute-tr-equiv-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X : UU l1} →
-  equiv-tr-equiv f id-equiv ＝ id-equiv
-compute-equiv-tr-equiv-id-equiv f {X} =
+  tr-equiv f id-equiv ＝ id-equiv
+compute-tr-equiv-id-equiv f {X} =
   (ap (equiv-tr f) (compute-eq-equiv-id-equiv X)) ∙ (equiv-tr-refl f)
 ```
 
 ### Transport along equivalences preserves composition of equivalences
 
 ```agda
-distributive-tr-equiv-equiv-comp :
+distributive-map-tr-equiv-equiv-comp :
   {l1 l2 : Level} (f : UU l1 → UU l2)
   {X Y Z : UU l1} (e : X ≃ Y) (e' : Y ≃ Z) →
-  tr-equiv f (e' ∘e e) ~ (tr-equiv f e' ∘ tr-equiv f e)
-distributive-tr-equiv-equiv-comp f {X} {Y} {Z} e e' x =
+  map-tr-equiv f (e' ∘e e) ~ (map-tr-equiv f e' ∘ map-tr-equiv f e)
+distributive-map-tr-equiv-equiv-comp f {X} {Y} {Z} e e' x =
   ( ap (λ p → tr f p x) (inv (compute-eq-equiv-equiv-comp X Y Z e e'))) ∙
   ( tr-concat (eq-equiv X Y e) (eq-equiv Y Z e') x)
 
-distributive-equiv-tr-equiv-equiv-comp :
+distributive-tr-equiv-equiv-comp :
   {l1 l2 : Level} (f : UU l1 → UU l2)
   {X Y Z : UU l1} (e : X ≃ Y) (e' : Y ≃ Z) →
-  equiv-tr-equiv f (e' ∘e e) ＝ (equiv-tr-equiv f e' ∘e equiv-tr-equiv f e)
-distributive-equiv-tr-equiv-equiv-comp f {X} {Y} {Z} e e' =
-  eq-htpy-equiv (distributive-tr-equiv-equiv-comp f e e')
+  tr-equiv f (e' ∘e e) ＝ (tr-equiv f e' ∘e tr-equiv f e)
+distributive-tr-equiv-equiv-comp f {X} {Y} {Z} e e' =
+  eq-htpy-equiv (distributive-map-tr-equiv-equiv-comp f e e')
 ```
 
 ### Transporting along an equivalence and its inverse is just the identity
 
 ```agda
-is-section-tr-equiv :
+is-section-map-tr-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2)
   {X Y : UU l1} (e : X ≃ Y) →
-  (tr-equiv f (inv-equiv e) ∘ tr-equiv f e) ~ id
-is-section-tr-equiv f {X} {Y} e x =
+  (map-tr-equiv f (inv-equiv e) ∘ map-tr-equiv f e) ~ id
+is-section-map-tr-equiv f {X} {Y} e x =
   ( ap
-    ( λ p → tr f p (tr-equiv f e x))
+    ( λ p → tr f p (map-tr-equiv f e x))
     ( inv (commutativity-inv-eq-equiv X Y e))) ∙
   ( is-section-tr f (eq-equiv X Y e) x)
 
-is-retraction-tr-equiv :
+is-retraction-map-tr-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2)
   {X Y : UU l1} (e : X ≃ Y) →
-  (tr-equiv f e ∘ tr-equiv f (inv-equiv e)) ~ id
-is-retraction-tr-equiv f {X} {Y} e x =
+  (map-tr-equiv f e ∘ map-tr-equiv f (inv-equiv e)) ~ id
+is-retraction-map-tr-equiv f {X} {Y} e x =
   ( ap
-    ( tr-equiv f e ∘ (λ p → tr f p x))
+    ( map-tr-equiv f e ∘ (λ p → tr f p x))
     ( inv (commutativity-inv-eq-equiv X Y e))) ∙
   ( is-retraction-tr f (eq-equiv X Y e) x)
 ```
@@ -128,63 +127,63 @@ is-retraction-tr-equiv f {X} {Y} e x =
 ### Transposing transport along the inverse of an equivalence
 
 ```agda
-eq-transpose-tr-equiv :
+eq-transpose-map-tr-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2)
   {X Y : UU l1} (e : X ≃ Y) {u : f X} {v : f Y} →
-  v ＝ tr-equiv f e u → tr-equiv f (inv-equiv e) v ＝ u
-eq-transpose-tr-equiv f e {u} p =
-  (ap (tr-equiv f (inv-equiv e)) p) ∙ (is-section-tr-equiv f e u)
+  v ＝ map-tr-equiv f e u → map-tr-equiv f (inv-equiv e) v ＝ u
+eq-transpose-map-tr-equiv f e {u} p =
+  (ap (map-tr-equiv f (inv-equiv e)) p) ∙ (is-section-map-tr-equiv f e u)
 
-eq-transpose-tr-equiv' :
+eq-transpose-map-tr-equiv' :
   {l1 l2 : Level} (f : UU l1 → UU l2)
   {X Y : UU l1} (e : X ≃ Y) {u : f X} {v : f Y} →
-  tr-equiv f e u ＝ v → u ＝ tr-equiv f (inv-equiv e) v
-eq-transpose-tr-equiv' f e {u} p =
-  (inv (is-section-tr-equiv f e u)) ∙ (ap (tr-equiv f (inv-equiv e)) p)
+  map-tr-equiv f e u ＝ v → u ＝ map-tr-equiv f (inv-equiv e) v
+eq-transpose-map-tr-equiv' f e {u} p =
+  (inv (is-section-map-tr-equiv f e u)) ∙ (ap (map-tr-equiv f (inv-equiv e)) p)
 ```
 
 ### Substitution law for transport along equivalences
 
 ```agda
-substitution-tr-equiv :
+substitution-map-tr-equiv :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2) {X Y : UU l1}
-  (e : X ≃ Y) → tr-equiv g (ap-equiv f e) ~ tr-equiv (g ∘ f) e
-substitution-tr-equiv g f {X} {Y} e X' =
-  ( ap (λ p → tr g p X') (is-retraction-eq-equiv (ap-eq-equiv f e))) ∙
+  (e : X ≃ Y) → map-tr-equiv g (ap-equiv f e) ~ map-tr-equiv (g ∘ f) e
+substitution-map-tr-equiv g f {X} {Y} e X' =
+  ( ap (λ p → tr g p X') (is-retraction-eq-equiv (eq-ap-equiv f e))) ∙
   ( substitution-tr g f (eq-equiv X Y e))
 
-substitution-equiv-tr-equiv :
+substitution-tr-equiv :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2) {X Y : UU l1}
-  (e : X ≃ Y) → equiv-tr-equiv g (ap-equiv f e) ＝ equiv-tr-equiv (g ∘ f) e
-substitution-equiv-tr-equiv g f e = eq-htpy-equiv (substitution-tr-equiv g f e)
+  (e : X ≃ Y) → tr-equiv g (ap-equiv f e) ＝ tr-equiv (g ∘ f) e
+substitution-tr-equiv g f e = eq-htpy-equiv (substitution-map-tr-equiv g f e)
 ```
 
 ### Transporting along the action on equivalences of a function
 
 ```agda
-compute-tr-equiv-ap-equiv :
+compute-map-tr-equiv-ap-equiv :
   {l1 l2 l3 l4 : Level} {B : UU l1 → UU l2} {D : UU l3 → UU l4}
   (f : UU l1 → UU l3) (g : (X : UU l1) → B X → D (f X))
   {X Y : UU l1} (e : X ≃ Y) (X' : B X) →
-  tr-equiv D (ap-equiv f e) (g X X') ＝ g Y (tr-equiv B e X')
-compute-tr-equiv-ap-equiv {D = D} f g {X} {Y} e X' =
-  ( ap (λ p → tr D p (g X X')) (is-retraction-eq-equiv (ap-eq-equiv f e))) ∙
+  map-tr-equiv D (ap-equiv f e) (g X X') ＝ g Y (map-tr-equiv B e X')
+compute-map-tr-equiv-ap-equiv {D = D} f g {X} {Y} e X' =
+  ( ap (λ p → tr D p (g X X')) (is-retraction-eq-equiv (eq-ap-equiv f e))) ∙
   ( tr-ap f g (eq-equiv X Y e) X')
 ```
 
 ### Transport along equivalences and the action on equivalences in the universe coincide
 
 ```agda
-eq-equiv-tr-equiv-ap-equiv :
+eq-tr-equiv-ap-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
-  (e : X ≃ Y) → equiv-tr-equiv f e ＝ ap-equiv f e
-eq-equiv-tr-equiv-ap-equiv f {X} =
-    ind-equiv X
-      ( λ Y e → equiv-tr-equiv f e ＝ ap-equiv f e)
-      ( compute-equiv-tr-equiv-id-equiv f ∙ inv (compute-ap-equiv-id-equiv f X))
+  (e : X ≃ Y) → tr-equiv f e ＝ ap-equiv f e
+eq-tr-equiv-ap-equiv f {X} =
+  ind-equiv X
+    ( λ Y e → tr-equiv f e ＝ ap-equiv f e)
+    ( compute-tr-equiv-id-equiv f ∙ inv (compute-ap-equiv-id-equiv f X))
 
-eq-tr-equiv-map-ap-equiv :
+eq-map-tr-equiv-map-ap-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
-  (e : X ≃ Y) → tr-equiv f e ＝ map-equiv (ap-equiv f e)
-eq-tr-equiv-map-ap-equiv f e = ap pr1 (eq-equiv-tr-equiv-ap-equiv f e)
+  (e : X ≃ Y) → map-tr-equiv f e ＝ map-ap-equiv f e
+eq-map-tr-equiv-map-ap-equiv f e = ap map-equiv (eq-tr-equiv-ap-equiv f e)
 ```

From 8f0c94fa91d14b02efe52260725a49b3d895dae0 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 28 Aug 2023 19:45:36 +0200
Subject: [PATCH 25/37] fix imports

---
 .../dependent-suspension-structures.lagda.md                    | 2 +-
 .../dependent-universal-property-suspensions.lagda.md           | 2 +-
 .../descent-circle-function-types.lagda.md                      | 2 +-
 src/synthetic-homotopy-theory/sections-descent-circle.lagda.md  | 2 +-
 src/synthetic-homotopy-theory/suspension-structures.lagda.md    | 2 +-
 .../suspensions-of-pointed-types.lagda.md                       | 2 +-
 .../universal-property-suspensions-of-pointed-types.lagda.md    | 2 +-
 .../universal-property-suspensions.lagda.md                     | 2 +-
 8 files changed, 8 insertions(+), 8 deletions(-)

diff --git a/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md b/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md
index a3ab28bd9a..33e7455434 100644
--- a/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md b/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md
index 8c6aa3c27e..0cf2e7b48b 100644
--- a/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md b/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md
index 15f9bbe907..4a56bcd36b 100644
--- a/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md
+++ b/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md
@@ -17,7 +17,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.functoriality-function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.descent-circle
diff --git a/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md b/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md
index af2782990c..d1c720d06b 100644
--- a/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md
@@ -23,7 +23,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.path-algebra
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.descent-circle
diff --git a/src/synthetic-homotopy-theory/suspension-structures.lagda.md b/src/synthetic-homotopy-theory/suspension-structures.lagda.md
index 4e3417d2a0..9a931276e9 100644
--- a/src/synthetic-homotopy-theory/suspension-structures.lagda.md
+++ b/src/synthetic-homotopy-theory/suspension-structures.lagda.md
@@ -24,7 +24,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md b/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md
index af23cf6315..66d7735858 100644
--- a/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md
+++ b/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md b/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
index a608a985a0..c3ca4487dd 100644
--- a/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md b/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md
index 3cd5f42732..4d9bbdad38 100644
--- a/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type

From a54e3281c6c2259e9e73db7c25edea1f7d37bcde Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 28 Aug 2023 21:17:47 +0200
Subject: [PATCH 26/37] fix links

---
 src/foundation/action-on-equivalences-subuniverses.lagda.md | 4 ++--
 src/synthetic-homotopy-theory/descent-circle.lagda.md       | 4 ++--
 2 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/foundation/action-on-equivalences-subuniverses.lagda.md b/src/foundation/action-on-equivalences-subuniverses.lagda.md
index 588b6c11be..bd9d0a6753 100644
--- a/src/foundation/action-on-equivalences-subuniverses.lagda.md
+++ b/src/foundation/action-on-equivalences-subuniverses.lagda.md
@@ -31,8 +31,8 @@ open import foundation-core.subtypes
 
 Given a [subuniverse](foundation.subuniverses.md) `P`, any family of types `B`
 indexed by types of `P` has an
-[action on equivalences](foundation.action-on-equivalences.md) obtained by using
-the [univalence axiom](foundation.univalence.md).
+[action on equivalences](foundation.action-on-equivalences-functions.md)
+obtained by using the [univalence axiom](foundation.univalence.md).
 
 ## Definition
 
diff --git a/src/synthetic-homotopy-theory/descent-circle.lagda.md b/src/synthetic-homotopy-theory/descent-circle.lagda.md
index a958ac25a5..0fd8cb8fbd 100644
--- a/src/synthetic-homotopy-theory/descent-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/descent-circle.lagda.md
@@ -121,8 +121,8 @@ module _
 ### Canonical descent data for a family over the circle
 
 A type family over the circle gives rise to its canonical descent data, obtained
-by evaluation at `base` and [transporting](foundation-core.transport.md) along
-`loop`.
+by evaluation at `base` and
+[transporting](foundation-core.transport-along-identifications.md) along `loop`.
 
 ```agda
 ev-descent-data-circle :

From 0ca6fbd19114974f63a12183dd9da3cbb9687710 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Wed, 6 Sep 2023 17:08:16 +0200
Subject: [PATCH 27/37] Revert `transport` -> `transport-along-identifications`

---
 .../bezouts-lemma-integers.lagda.md                   |  2 +-
 .../bezouts-lemma-natural-numbers.lagda.md            |  2 +-
 src/elementary-number-theory/cofibonacci.lagda.md     |  2 +-
 .../congruence-integers.lagda.md                      |  2 +-
 .../congruence-natural-numbers.lagda.md               |  2 +-
 .../distance-natural-numbers.lagda.md                 |  2 +-
 .../divisibility-integers.lagda.md                    |  2 +-
 .../divisibility-natural-numbers.lagda.md             |  2 +-
 .../fibonacci-sequence.lagda.md                       |  2 +-
 .../fundamental-theorem-of-arithmetic.lagda.md        |  2 +-
 .../greatest-common-divisor-integers.lagda.md         |  2 +-
 .../greatest-common-divisor-natural-numbers.lagda.md  |  2 +-
 .../inequality-integers.lagda.md                      |  2 +-
 src/elementary-number-theory/powers-of-two.lagda.md   |  2 +-
 src/elementary-number-theory/prime-numbers.lagda.md   |  2 +-
 .../reduced-integer-fractions.lagda.md                |  2 +-
 .../relatively-prime-natural-numbers.lagda.md         |  2 +-
 .../strict-inequality-natural-numbers.lagda.md        |  2 +-
 ...-ordering-principle-standard-finite-types.lagda.md |  2 +-
 .../cartier-delooping-sign-homomorphism.lagda.md      |  2 +-
 .../concrete-quaternion-group.lagda.md                |  2 +-
 .../delooping-sign-homomorphism.lagda.md              |  2 +-
 src/finite-group-theory/orbits-permutations.lagda.md  |  2 +-
 .../simpson-delooping-sign-homomorphism.lagda.md      |  2 +-
 src/finite-group-theory/transpositions.lagda.md       |  2 +-
 src/foundation-core.lagda.md                          |  2 +-
 src/foundation-core/contractible-types.lagda.md       |  2 +-
 .../dependent-identifications.lagda.md                | 11 +++++------
 .../equality-dependent-pair-types.lagda.md            |  2 +-
 src/foundation-core/fibers-of-maps.lagda.md           |  2 +-
 .../functoriality-dependent-function-types.lagda.md   |  2 +-
 .../functoriality-dependent-pair-types.lagda.md       |  2 +-
 src/foundation-core/homotopies.lagda.md               |  2 +-
 src/foundation-core/identity-types.lagda.md           |  4 ++--
 src/foundation-core/propositions.lagda.md             |  2 +-
 src/foundation-core/singleton-induction.lagda.md      |  2 +-
 src/foundation-core/small-types.lagda.md              |  2 +-
 src/foundation-core/subtypes.lagda.md                 |  2 +-
 ...ng-identifications.lagda.md => transport.lagda.md} |  5 ++---
 src/foundation-core/truncated-types.lagda.md          |  2 +-
 src/foundation-core/univalence.lagda.md               |  2 +-
 src/foundation.lagda.md                               |  2 +-
 .../action-on-equivalences-functions.lagda.md         |  3 +--
 .../action-on-equivalences-subuniverses.lagda.md      |  2 +-
 ...on-on-identifications-dependent-functions.lagda.md |  6 +++---
 .../action-on-identifications-functions.lagda.md      |  2 +-
 ...ic-law-coproduct-and-sigma-decompositions.lagda.md |  2 +-
 ...thmetic-law-product-and-pi-decompositions.lagda.md |  2 +-
 src/foundation/binary-transport.lagda.md              |  2 +-
 src/foundation/cartesian-product-types.lagda.md       |  2 +-
 src/foundation/cones-over-cospans.lagda.md            |  2 +-
 src/foundation/coproduct-decompositions.lagda.md      |  2 +-
 src/foundation/decidable-equality.lagda.md            |  2 +-
 .../decidable-equivalence-relations.lagda.md          |  2 +-
 src/foundation/decidable-propositions.lagda.md        |  2 +-
 src/foundation/decidable-subtypes.lagda.md            |  2 +-
 src/foundation/dependent-binomial-theorem.lagda.md    |  2 +-
 src/foundation/dependent-identifications.lagda.md     |  2 +-
 src/foundation/double-negation-modality.lagda.md      |  2 +-
 .../equality-cartesian-product-types.lagda.md         |  2 +-
 src/foundation/equality-dependent-pair-types.lagda.md |  2 +-
 src/foundation/equality-fibers-of-maps.lagda.md       |  2 +-
 src/foundation/equivalence-relations.lagda.md         |  2 +-
 src/foundation/exclusive-disjunction.lagda.md         |  2 +-
 src/foundation/fiber-inclusions.lagda.md              |  2 +-
 src/foundation/full-subtypes.lagda.md                 |  2 +-
 src/foundation/function-types.lagda.md                |  2 +-
 src/foundation/functoriality-coproduct-types.lagda.md |  2 +-
 .../functoriality-dependent-function-types.lagda.md   |  2 +-
 .../functoriality-dependent-pair-types.lagda.md       |  2 +-
 src/foundation/homotopies.lagda.md                    |  2 +-
 src/foundation/identity-systems.lagda.md              |  2 +-
 src/foundation/identity-types.lagda.md                |  6 +++---
 ...uction-principle-propositional-truncation.lagda.md |  2 +-
 src/foundation/isolated-points.lagda.md               |  2 +-
 src/foundation/locally-small-types.lagda.md           |  2 +-
 src/foundation/partitions.lagda.md                    |  2 +-
 src/foundation/perfect-images.lagda.md                |  2 +-
 .../pointed-torsorial-type-families.lagda.md          |  2 +-
 .../products-unordered-pairs-of-types.lagda.md        |  2 +-
 src/foundation/propositional-extensionality.lagda.md  |  2 +-
 src/foundation/propositional-truncations.lagda.md     |  2 +-
 src/foundation/pullbacks.lagda.md                     |  2 +-
 src/foundation/singleton-induction.lagda.md           |  2 +-
 src/foundation/structure.lagda.md                     |  2 +-
 src/foundation/subuniverses.lagda.md                  |  2 +-
 src/foundation/symmetric-difference.lagda.md          |  2 +-
 src/foundation/transport-along-equivalences.lagda.md  |  7 +++----
 ...ng-identifications.lagda.md => transport.lagda.md} |  9 ++++-----
 .../type-theoretic-principle-of-choice.lagda.md       |  2 +-
 ...nivalence-implies-function-extensionality.lagda.md |  2 +-
 src/foundation/univalent-type-families.lagda.md       |  2 +-
 src/foundation/universal-property-image.lagda.md      |  2 +-
 .../universal-property-set-quotients.lagda.md         |  2 +-
 .../equivalences-enriched-undirected-graphs.lagda.md  |  2 +-
 .../equivalences-undirected-graphs.lagda.md           |  2 +-
 src/graph-theory/fibers-directed-graphs.lagda.md      |  2 +-
 src/graph-theory/morphisms-undirected-graphs.lagda.md |  2 +-
 src/graph-theory/undirected-graphs.lagda.md           |  2 +-
 src/graph-theory/walks-directed-graphs.lagda.md       |  2 +-
 src/group-theory/concrete-group-actions.lagda.md      |  2 +-
 .../homomorphisms-concrete-group-actions.lagda.md     |  2 +-
 src/group-theory/normal-subgroups.lagda.md            |  2 +-
 src/group-theory/subgroups.lagda.md                   |  2 +-
 src/group-theory/torsors.lagda.md                     |  2 +-
 .../transitive-concrete-group-actions.lagda.md        |  2 +-
 src/higher-group-theory/higher-group-actions.lagda.md |  2 +-
 .../homomorphisms-higher-group-actions.lagda.md       |  2 +-
 src/linear-algebra/vectors.lagda.md                   |  2 +-
 src/lists/functoriality-lists.lagda.md                |  2 +-
 src/lists/permutation-lists.lagda.md                  |  2 +-
 src/lists/permutation-vectors.lagda.md                |  2 +-
 src/lists/sort-by-insertion-vectors.lagda.md          |  2 +-
 src/order-theory/finitely-graded-posets.lagda.md      |  2 +-
 src/organic-chemistry/ethane.lagda.md                 |  2 +-
 .../extensions-of-maps.lagda.md                       |  2 +-
 .../function-classes.lagda.md                         |  2 +-
 .../open-modalities.lagda.md                          |  2 +-
 src/polytopes/abstract-polytopes.lagda.md             |  2 +-
 .../ideals-generated-by-subsets-rings.lagda.md        |  2 +-
 .../left-ideals-generated-by-subsets-rings.lagda.md   |  2 +-
 .../right-ideals-generated-by-subsets-rings.lagda.md  |  2 +-
 src/set-theory/cumulative-hierarchy.lagda.md          |  2 +-
 ...position-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...nentials-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...products-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...nentials-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...products-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...position-species-of-types-in-subuniverses.lagda.md |  2 +-
 src/species/species-of-types-in-subuniverses.lagda.md |  2 +-
 src/species/species-of-types.lagda.md                 |  2 +-
 src/structured-types/pointed-maps.lagda.md            |  2 +-
 src/synthetic-homotopy-theory/26-descent.lagda.md     |  2 +-
 src/synthetic-homotopy-theory/26-id-pushout.lagda.md  |  2 +-
 src/synthetic-homotopy-theory/circle.lagda.md         |  2 +-
 .../dependent-cocones-under-spans.lagda.md            |  2 +-
 .../dependent-pullback-property-pushouts.lagda.md     |  2 +-
 .../dependent-suspension-structures.lagda.md          |  2 +-
 .../dependent-universal-property-pushouts.lagda.md    |  2 +-
 .../dependent-universal-property-suspensions.lagda.md |  2 +-
 .../descent-circle-function-types.lagda.md            |  2 +-
 src/synthetic-homotopy-theory/descent-circle.lagda.md |  6 +++---
 src/synthetic-homotopy-theory/free-loops.lagda.md     |  2 +-
 src/synthetic-homotopy-theory/interval-type.lagda.md  |  2 +-
 .../multiplication-circle.lagda.md                    |  2 +-
 .../sections-descent-circle.lagda.md                  |  2 +-
 .../suspension-structures.lagda.md                    |  2 +-
 .../suspensions-of-pointed-types.lagda.md             |  2 +-
 .../suspensions-of-types.lagda.md                     |  2 +-
 .../universal-cover-circle.lagda.md                   |  2 +-
 .../universal-property-circle.lagda.md                |  2 +-
 ...sal-property-suspensions-of-pointed-types.lagda.md |  2 +-
 .../universal-property-suspensions.lagda.md           |  2 +-
 src/trees/bases-directed-trees.lagda.md               |  2 +-
 src/trees/directed-trees.lagda.md                     |  2 +-
 src/trees/elementhood-relation-w-types.lagda.md       |  2 +-
 src/trees/enriched-directed-trees.lagda.md            |  2 +-
 src/trees/equivalences-directed-trees.lagda.md        |  2 +-
 .../equivalences-enriched-directed-trees.lagda.md     |  2 +-
 src/trees/extensional-w-types.lagda.md                |  2 +-
 .../functoriality-combinator-directed-trees.lagda.md  |  2 +-
 src/trees/functoriality-w-types.lagda.md              |  2 +-
 src/trees/induction-w-types.lagda.md                  |  2 +-
 src/trees/morphisms-directed-trees.lagda.md           |  2 +-
 src/trees/morphisms-enriched-directed-trees.lagda.md  |  2 +-
 src/trees/polynomial-endofunctors.lagda.md            |  2 +-
 src/trees/ranks-of-elements-w-types.lagda.md          |  2 +-
 src/trees/rooted-morphisms-directed-trees.lagda.md    |  2 +-
 .../rooted-morphisms-enriched-directed-trees.lagda.md |  2 +-
 src/trees/small-multisets.lagda.md                    |  2 +-
 ...ements-coalgebras-polynomial-endofunctors.lagda.md |  2 +-
 .../underlying-trees-of-elements-of-w-types.lagda.md  |  2 +-
 src/trees/undirected-trees.lagda.md                   |  2 +-
 src/trees/universal-multiset.lagda.md                 |  2 +-
 src/trees/w-types.lagda.md                            |  2 +-
 src/type-theories/dependent-type-theories.lagda.md    |  2 +-
 .../fibered-dependent-type-theories.lagda.md          |  2 +-
 .../sections-dependent-type-theories.lagda.md         |  2 +-
 src/type-theories/simple-type-theories.lagda.md       |  2 +-
 .../2-element-decidable-subtypes.lagda.md             |  2 +-
 .../2-element-subtypes.lagda.md                       |  2 +-
 src/univalent-combinatorics/2-element-types.lagda.md  |  2 +-
 .../counting-dependent-pair-types.lagda.md            |  2 +-
 ...cycle-prime-decomposition-natural-numbers.lagda.md |  2 +-
 .../decidable-subtypes.lagda.md                       |  2 +-
 .../dependent-pair-types.lagda.md                     |  2 +-
 .../equivalences-cubes.lagda.md                       |  2 +-
 src/univalent-combinatorics/fibers-of-maps.lagda.md   |  2 +-
 src/univalent-combinatorics/finite-types.lagda.md     |  2 +-
 .../orientations-complete-undirected-graph.lagda.md   |  2 +-
 src/univalent-combinatorics/pi-finite-types.lagda.md  |  2 +-
 src/universal-algebra/terms-over-signatures.lagda.md  |  2 +-
 192 files changed, 209 insertions(+), 214 deletions(-)
 rename src/foundation-core/{transport-along-identifications.lagda.md => transport.lagda.md} (90%)
 rename src/foundation/{transport-along-identifications.lagda.md => transport.lagda.md} (94%)

diff --git a/src/elementary-number-theory/bezouts-lemma-integers.lagda.md b/src/elementary-number-theory/bezouts-lemma-integers.lagda.md
index b94ea9b729..536b969aa3 100644
--- a/src/elementary-number-theory/bezouts-lemma-integers.lagda.md
+++ b/src/elementary-number-theory/bezouts-lemma-integers.lagda.md
@@ -28,7 +28,7 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 ```
 
diff --git a/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md b/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md
index ad127678d2..19ad5b7dc7 100755
--- a/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md
@@ -37,7 +37,7 @@ open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import univalent-combinatorics.standard-finite-types
diff --git a/src/elementary-number-theory/cofibonacci.lagda.md b/src/elementary-number-theory/cofibonacci.lagda.md
index 8d1f91db9b..e80bd5387e 100644
--- a/src/elementary-number-theory/cofibonacci.lagda.md
+++ b/src/elementary-number-theory/cofibonacci.lagda.md
@@ -22,7 +22,7 @@ open import foundation.decidable-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/congruence-integers.lagda.md b/src/elementary-number-theory/congruence-integers.lagda.md
index 9844348d01..51d82b086d 100644
--- a/src/elementary-number-theory/congruence-integers.lagda.md
+++ b/src/elementary-number-theory/congruence-integers.lagda.md
@@ -22,7 +22,7 @@ open import foundation.binary-relations
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/congruence-natural-numbers.lagda.md b/src/elementary-number-theory/congruence-natural-numbers.lagda.md
index 57781d53a3..732c75a31d 100644
--- a/src/elementary-number-theory/congruence-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/congruence-natural-numbers.lagda.md
@@ -19,7 +19,7 @@ open import foundation.binary-relations
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import univalent-combinatorics.standard-finite-types
diff --git a/src/elementary-number-theory/distance-natural-numbers.lagda.md b/src/elementary-number-theory/distance-natural-numbers.lagda.md
index 0426593e35..4aa61b2b9e 100644
--- a/src/elementary-number-theory/distance-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/distance-natural-numbers.lagda.md
@@ -19,7 +19,7 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/divisibility-integers.lagda.md b/src/elementary-number-theory/divisibility-integers.lagda.md
index b6c1fc7453..e570a1a875 100644
--- a/src/elementary-number-theory/divisibility-integers.lagda.md
+++ b/src/elementary-number-theory/divisibility-integers.lagda.md
@@ -27,7 +27,7 @@ open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositional-maps
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md
index 869a1a4189..97ce7e55af 100644
--- a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md
@@ -23,7 +23,7 @@ open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-maps
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/elementary-number-theory/fibonacci-sequence.lagda.md b/src/elementary-number-theory/fibonacci-sequence.lagda.md
index 634cc34706..ed4d4e50cf 100644
--- a/src/elementary-number-theory/fibonacci-sequence.lagda.md
+++ b/src/elementary-number-theory/fibonacci-sequence.lagda.md
@@ -17,7 +17,7 @@ open import elementary-number-theory.relatively-prime-natural-numbers
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 ```
 
 </details>
diff --git a/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md b/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md
index 992aa4a2fa..b05cff6d00 100644
--- a/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md
+++ b/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md
@@ -35,7 +35,7 @@ open import foundation.empty-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md b/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md
index 2dc2e9236a..8fa61c143e 100644
--- a/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md
+++ b/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md
@@ -24,7 +24,7 @@ open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md b/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md
index 8b4fdaaa35..32b773f17b 100644
--- a/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md
@@ -30,7 +30,7 @@ open import foundation.empty-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/inequality-integers.lagda.md b/src/elementary-number-theory/inequality-integers.lagda.md
index aa92163af9..2955f6bf44 100644
--- a/src/elementary-number-theory/inequality-integers.lagda.md
+++ b/src/elementary-number-theory/inequality-integers.lagda.md
@@ -19,7 +19,7 @@ open import foundation.function-types
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/powers-of-two.lagda.md b/src/elementary-number-theory/powers-of-two.lagda.md
index 9efb3ca6ba..d1b2c00db1 100644
--- a/src/elementary-number-theory/powers-of-two.lagda.md
+++ b/src/elementary-number-theory/powers-of-two.lagda.md
@@ -21,7 +21,7 @@ open import foundation.dependent-pair-types
 open import foundation.equality-cartesian-product-types
 open import foundation.function-types
 open import foundation.split-surjective-maps
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import foundation-core.cartesian-product-types
diff --git a/src/elementary-number-theory/prime-numbers.lagda.md b/src/elementary-number-theory/prime-numbers.lagda.md
index 00cb8b2065..ad4ce088ba 100644
--- a/src/elementary-number-theory/prime-numbers.lagda.md
+++ b/src/elementary-number-theory/prime-numbers.lagda.md
@@ -28,7 +28,7 @@ open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/reduced-integer-fractions.lagda.md b/src/elementary-number-theory/reduced-integer-fractions.lagda.md
index 06aa1843e5..a6dd71808f 100644
--- a/src/elementary-number-theory/reduced-integer-fractions.lagda.md
+++ b/src/elementary-number-theory/reduced-integer-fractions.lagda.md
@@ -26,7 +26,7 @@ open import foundation.equality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md b/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md
index fbf54cb39d..61c91bb897 100644
--- a/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md
@@ -20,7 +20,7 @@ open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md
index a70e22d17a..847e67d95c 100644
--- a/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md
@@ -22,7 +22,7 @@ open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md b/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md
index 3e7b58ad9d..e392a72dba 100644
--- a/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md
+++ b/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md
@@ -31,7 +31,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-empty-type
 open import foundation.type-arithmetic-unit-type
diff --git a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
index c342c02201..5da4a8f958 100644
--- a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
@@ -28,7 +28,7 @@ open import foundation.mere-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/finite-group-theory/concrete-quaternion-group.lagda.md b/src/finite-group-theory/concrete-quaternion-group.lagda.md
index d03507a4e9..0c66d78790 100644
--- a/src/finite-group-theory/concrete-quaternion-group.lagda.md
+++ b/src/finite-group-theory/concrete-quaternion-group.lagda.md
@@ -13,7 +13,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.isolated-points
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import univalent-combinatorics.complements-isolated-points
diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index c00d5d7b7f..a4c900d575 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -48,7 +48,7 @@ open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.reflecting-maps-equivalence-relations
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.truncated-types
 open import foundation.uniqueness-set-quotients
 open import foundation.unit-type
diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md
index 5e5fef1f25..0d3195117e 100644
--- a/src/finite-group-theory/orbits-permutations.lagda.md
+++ b/src/finite-group-theory/orbits-permutations.lagda.md
@@ -48,7 +48,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.repetitions-of-values
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
index 64358bec43..8ed74b4046 100644
--- a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
@@ -42,7 +42,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md
index 06b4c99881..02d1b40715 100644
--- a/src/finite-group-theory/transpositions.lagda.md
+++ b/src/finite-group-theory/transpositions.lagda.md
@@ -40,7 +40,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/foundation-core.lagda.md b/src/foundation-core.lagda.md
index 8298257aca..72aaa8803c 100644
--- a/src/foundation-core.lagda.md
+++ b/src/foundation-core.lagda.md
@@ -48,7 +48,7 @@ open import foundation-core.sets public
 open import foundation-core.singleton-induction public
 open import foundation-core.small-types public
 open import foundation-core.subtypes public
-open import foundation-core.transport-along-identifications public
+open import foundation-core.transport public
 open import foundation-core.truncated-maps public
 open import foundation-core.truncated-types public
 open import foundation-core.truncation-levels public
diff --git a/src/foundation-core/contractible-types.lagda.md b/src/foundation-core/contractible-types.lagda.md
index 73fc88a726..cfa56145e5 100644
--- a/src/foundation-core/contractible-types.lagda.md
+++ b/src/foundation-core/contractible-types.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.equality-dependent-pair-types
 open import foundation-core.equivalences
 open import foundation-core.identity-types
 open import foundation-core.retractions
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation-core/dependent-identifications.lagda.md b/src/foundation-core/dependent-identifications.lagda.md
index 55178798aa..9a5da82082 100644
--- a/src/foundation-core/dependent-identifications.lagda.md
+++ b/src/foundation-core/dependent-identifications.lagda.md
@@ -10,7 +10,7 @@ module foundation-core.dependent-identifications where
 open import foundation.universe-levels
 
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
@@ -26,11 +26,10 @@ identification
   tr B p u ＝ v,
 ```
 
-where `tr` is the
-[transport](foundation-core.transport-along-identifications.md) function.
-Dependent identifications also satisfy groupoid laws, which are formulated
-appropriately as dependent identifications. The groupoid laws for dependent
-identifications are proven in
+where `tr` is the [transport](foundation-core.transport.md) function. Dependent
+identifications also satisfy groupoid laws, which are formulated appropriately
+as dependent identifications. The groupoid laws for dependent identifications
+are proven in
 [`foundation.dependent-identifications`](foundation.dependent-identifications.md).
 
 ## Definition
diff --git a/src/foundation-core/equality-dependent-pair-types.lagda.md b/src/foundation-core/equality-dependent-pair-types.lagda.md
index f07acb266f..d9b33e2faa 100644
--- a/src/foundation-core/equality-dependent-pair-types.lagda.md
+++ b/src/foundation-core/equality-dependent-pair-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation-core/fibers-of-maps.lagda.md b/src/foundation-core/fibers-of-maps.lagda.md
index 5d5c22dbf4..8cffa5184c 100644
--- a/src/foundation-core/fibers-of-maps.lagda.md
+++ b/src/foundation-core/fibers-of-maps.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation-core/functoriality-dependent-function-types.lagda.md b/src/foundation-core/functoriality-dependent-function-types.lagda.md
index a48e86bd2e..33ff55f232 100644
--- a/src/foundation-core/functoriality-dependent-function-types.lagda.md
+++ b/src/foundation-core/functoriality-dependent-function-types.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.path-split-maps
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation-core/functoriality-dependent-pair-types.lagda.md b/src/foundation-core/functoriality-dependent-pair-types.lagda.md
index 8bf0dca52a..ada200ac6c 100644
--- a/src/foundation-core/functoriality-dependent-pair-types.lagda.md
+++ b/src/foundation-core/functoriality-dependent-pair-types.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.fibers-of-maps
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation-core/homotopies.lagda.md b/src/foundation-core/homotopies.lagda.md
index f262a3684d..d94bb57a82 100644
--- a/src/foundation-core/homotopies.lagda.md
+++ b/src/foundation-core/homotopies.lagda.md
@@ -15,7 +15,7 @@ open import foundation.universe-levels
 open import foundation-core.dependent-identifications
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation-core/identity-types.lagda.md b/src/foundation-core/identity-types.lagda.md
index 8e249858e9..0303ae6e3c 100644
--- a/src/foundation-core/identity-types.lagda.md
+++ b/src/foundation-core/identity-types.lagda.md
@@ -67,8 +67,8 @@ identifications in arbitrary types.
 | Path algebra                                      | [`foundation.path-algebra`](foundation.path-algebra.md)                                                                   |
 | Symmetric identity types                          | [`foundation.symmetric-identity-types`](foundation.symmetric-identity-types.md)                                           |
 | Torsorial type families                           | [`foundation.torsorial-type-families`](foundation.torsorial-type-families.md)                                             |
-| Transport along identifications (foundation)      | [`foundation.transport-along-identifications`](foundation.transport-along-identifications.md)                             |
-| Transport along identifications (foundation-core) | [`foundation-core.transport-along-identifications`](foundation-core.transport-along-identifications.md)                   |
+| Transport along identifications (foundation)      | [`foundation.transport`](foundation.transport.md)                                                                         |
+| Transport along identifications (foundation-core) | [`foundation-core.transport`](foundation-core.transport.md)                                                               |
 | The universal property of identity types          | [`foundation.universal-property-identity-types`](foundation.universal-property-identity-types.md)                         |
 
 ## Definition
diff --git a/src/foundation-core/propositions.lagda.md b/src/foundation-core/propositions.lagda.md
index 9901fae32b..45a5b25af2 100644
--- a/src/foundation-core/propositions.lagda.md
+++ b/src/foundation-core/propositions.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.function-extensionality
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation-core/singleton-induction.lagda.md b/src/foundation-core/singleton-induction.lagda.md
index 636f391214..aa772e61a2 100644
--- a/src/foundation-core/singleton-induction.lagda.md
+++ b/src/foundation-core/singleton-induction.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.sections
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation-core/small-types.lagda.md b/src/foundation-core/small-types.lagda.md
index 6f676a6604..0c01b6c305 100644
--- a/src/foundation-core/small-types.lagda.md
+++ b/src/foundation-core/small-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation.identity-types
 open import foundation.mere-equivalences
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/foundation-core/subtypes.lagda.md b/src/foundation-core/subtypes.lagda.md
index 523a820842..f738fedc79 100644
--- a/src/foundation-core/subtypes.lagda.md
+++ b/src/foundation-core/subtypes.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.logical-equivalences
 open import foundation-core.propositional-maps
 open import foundation-core.propositions
 open import foundation-core.sets
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation-core/transport-along-identifications.lagda.md b/src/foundation-core/transport.lagda.md
similarity index 90%
rename from src/foundation-core/transport-along-identifications.lagda.md
rename to src/foundation-core/transport.lagda.md
index 176a78d957..751cc379d2 100644
--- a/src/foundation-core/transport-along-identifications.lagda.md
+++ b/src/foundation-core/transport.lagda.md
@@ -1,7 +1,7 @@
 # Transport along identifications
 
 ```agda
-module foundation-core.transport-along-identifications where
+module foundation-core.transport where
 ```
 
 <details><summary>Imports</summary>
@@ -22,8 +22,7 @@ element `b : B x`, we can **transport** the element `b` along the identification
 `p` to obtain an element `tr B p b : B y`.
 
 The fact that `tr B p` is an [equivalence](foundation-core.equivalences.md) is
-recorded in
-[`foundation.transport-along-identifications`](foundation.transport-along-identifications.md).
+recorded in [`foundation.transport`](foundation.transport.md).
 
 ## Definition
 
diff --git a/src/foundation-core/truncated-types.lagda.md b/src/foundation-core/truncated-types.lagda.md
index 921e8c54f6..77b40c7c75 100644
--- a/src/foundation-core/truncated-types.lagda.md
+++ b/src/foundation-core/truncated-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.retractions
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 open import foundation-core.truncation-levels
 ```
 
diff --git a/src/foundation-core/univalence.lagda.md b/src/foundation-core/univalence.lagda.md
index 1192ecd6e2..eb72470b0f 100644
--- a/src/foundation-core/univalence.lagda.md
+++ b/src/foundation-core/univalence.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.contractible-types
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index 84e6c7532b..4622233d3d 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -255,8 +255,8 @@ open import foundation.symmetric-identity-types public
 open import foundation.symmetric-operations public
 open import foundation.tight-apartness-relations public
 open import foundation.torsorial-type-families public
+open import foundation.transport public
 open import foundation.transport-along-equivalences public
-open import foundation.transport-along-identifications public
 open import foundation.trivial-relaxed-sigma-decompositions public
 open import foundation.trivial-sigma-decompositions public
 open import foundation.truncated-equality public
diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index a28d652594..9ff204868d 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -29,8 +29,7 @@ to [identifications](foundation-core.identity-types.md) arising from the
 equivalences**.
 
 Alternatively, one can apply
-[transport along identifications](foundation-core.transport-along-identifications.md)
-to get
+[transport along identifications](foundation-core.transport.md) to get
 [transport along equivalences](foundation.transport-along-equivalences.md), but
 luckily, these two notions coincide.
 
diff --git a/src/foundation/action-on-equivalences-subuniverses.lagda.md b/src/foundation/action-on-equivalences-subuniverses.lagda.md
index bd9d0a6753..b9f13a1e86 100644
--- a/src/foundation/action-on-equivalences-subuniverses.lagda.md
+++ b/src/foundation/action-on-equivalences-subuniverses.lagda.md
@@ -12,7 +12,7 @@ open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.sets
 open import foundation.subuniverses
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/foundation/action-on-identifications-dependent-functions.lagda.md b/src/foundation/action-on-identifications-dependent-functions.lagda.md
index 6db5aec11d..e639d4d709 100644
--- a/src/foundation/action-on-identifications-dependent-functions.lagda.md
+++ b/src/foundation/action-on-identifications-dependent-functions.lagda.md
@@ -21,9 +21,9 @@ Given a dependent function `f : (x : A) → B x` and an
 [identification](foundation-core.identity-types.md) `p : x ＝ y` in `A`, we
 cannot directly compare the values `f x` and `f y`, since `f x` is an element of
 type `B x` while `f y` is an element of type `B y`. However, we can
-[transport](foundation-core.transport-along-identifications.md) the value `f x`
-along `p` to obtain an element of type `B y` that is comparable to the value
-`f y`. In other words, we can consider the type of
+[transport](foundation-core.transport.md) the value `f x` along `p` to obtain an
+element of type `B y` that is comparable to the value `f y`. In other words, we
+can consider the type of
 [dependent identifications](foundation-core.dependent-identifications.md) over
 `p` from `f x` to `f y`. The **dependent action on identifications** of `f` on
 `p` is a dependent identification
diff --git a/src/foundation/action-on-identifications-functions.lagda.md b/src/foundation/action-on-identifications-functions.lagda.md
index 4fd28c92ed..7ff1b0867d 100644
--- a/src/foundation/action-on-identifications-functions.lagda.md
+++ b/src/foundation/action-on-identifications-functions.lagda.md
@@ -12,7 +12,7 @@ open import foundation.universe-levels
 open import foundation-core.constant-maps
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md b/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md
index 225b0ebaa9..8e76432c07 100644
--- a/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md
+++ b/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md b/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md
index 0d028fa529..5a34d34c97 100644
--- a/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md
+++ b/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/binary-transport.lagda.md b/src/foundation/binary-transport.lagda.md
index 5ac3a40225..5de9ca3d4f 100644
--- a/src/foundation/binary-transport.lagda.md
+++ b/src/foundation/binary-transport.lagda.md
@@ -14,7 +14,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/cartesian-product-types.lagda.md b/src/foundation/cartesian-product-types.lagda.md
index 1c324495a7..eefc3e3266 100644
--- a/src/foundation/cartesian-product-types.lagda.md
+++ b/src/foundation/cartesian-product-types.lagda.md
@@ -14,7 +14,7 @@ open import foundation.subuniverses
 open import foundation.universe-levels
 
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/cones-over-cospans.lagda.md b/src/foundation/cones-over-cospans.lagda.md
index dcb1e94c20..40e0417706 100644
--- a/src/foundation/cones-over-cospans.lagda.md
+++ b/src/foundation/cones-over-cospans.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.contractible-types
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/coproduct-decompositions.lagda.md b/src/foundation/coproduct-decompositions.lagda.md
index 3bdb3d1749..f3219be2d7 100644
--- a/src/foundation/coproduct-decompositions.lagda.md
+++ b/src/foundation/coproduct-decompositions.lagda.md
@@ -32,7 +32,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.propositions
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 
 open import univalent-combinatorics.equality-standard-finite-types
 open import univalent-combinatorics.standard-finite-types
diff --git a/src/foundation/decidable-equality.lagda.md b/src/foundation/decidable-equality.lagda.md
index fd89e09b86..7e7f86ce21 100644
--- a/src/foundation/decidable-equality.lagda.md
+++ b/src/foundation/decidable-equality.lagda.md
@@ -28,7 +28,7 @@ open import foundation-core.injective-maps
 open import foundation-core.propositions
 open import foundation-core.retractions
 open import foundation-core.sets
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/decidable-equivalence-relations.lagda.md b/src/foundation/decidable-equivalence-relations.lagda.md
index e61f0003d3..6a2bd8bc09 100644
--- a/src/foundation/decidable-equivalence-relations.lagda.md
+++ b/src/foundation/decidable-equivalence-relations.lagda.md
@@ -45,7 +45,7 @@ open import foundation-core.identity-types
 open import foundation-core.logical-equivalences
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/decidable-propositions.lagda.md b/src/foundation/decidable-propositions.lagda.md
index baae1d24a5..6bed5895cd 100644
--- a/src/foundation/decidable-propositions.lagda.md
+++ b/src/foundation/decidable-propositions.lagda.md
@@ -34,7 +34,7 @@ open import foundation-core.propositions
 open import foundation-core.sets
 open import foundation-core.small-types
 open import foundation-core.subtypes
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 
 open import univalent-combinatorics.counting
 open import univalent-combinatorics.finite-types
diff --git a/src/foundation/decidable-subtypes.lagda.md b/src/foundation/decidable-subtypes.lagda.md
index e4700eb666..611ae47d43 100644
--- a/src/foundation/decidable-subtypes.lagda.md
+++ b/src/foundation/decidable-subtypes.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.identity-types
 open import foundation-core.injective-maps
 open import foundation-core.logical-equivalences
 open import foundation-core.propositions
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation/dependent-binomial-theorem.lagda.md b/src/foundation/dependent-binomial-theorem.lagda.md
index f6e388bcb0..ccd5203beb 100644
--- a/src/foundation/dependent-binomial-theorem.lagda.md
+++ b/src/foundation/dependent-binomial-theorem.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-dependent-function-types
 open import foundation.raising-universe-levels
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.universal-property-coproduct-types
 open import foundation.universal-property-dependent-pair-types
diff --git a/src/foundation/dependent-identifications.lagda.md b/src/foundation/dependent-identifications.lagda.md
index 50838bcafc..a194b8477e 100644
--- a/src/foundation/dependent-identifications.lagda.md
+++ b/src/foundation/dependent-identifications.lagda.md
@@ -10,7 +10,7 @@ open import foundation-core.dependent-identifications public
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import foundation-core.equivalences
diff --git a/src/foundation/double-negation-modality.lagda.md b/src/foundation/double-negation-modality.lagda.md
index ee2407d3ea..8420266ab1 100644
--- a/src/foundation/double-negation-modality.lagda.md
+++ b/src/foundation/double-negation-modality.lagda.md
@@ -12,7 +12,7 @@ open import foundation.universe-levels
 
 open import foundation-core.function-types
 open import foundation-core.propositions
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 
 open import orthogonal-factorization-systems.local-types
 open import orthogonal-factorization-systems.modal-operators
diff --git a/src/foundation/equality-cartesian-product-types.lagda.md b/src/foundation/equality-cartesian-product-types.lagda.md
index 7e385c0e60..5a4a6c6137 100644
--- a/src/foundation/equality-cartesian-product-types.lagda.md
+++ b/src/foundation/equality-cartesian-product-types.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/equality-dependent-pair-types.lagda.md b/src/foundation/equality-dependent-pair-types.lagda.md
index 426a1c39da..0de1851187 100644
--- a/src/foundation/equality-dependent-pair-types.lagda.md
+++ b/src/foundation/equality-dependent-pair-types.lagda.md
@@ -13,7 +13,7 @@ open import foundation.action-on-identifications-dependent-functions
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-identifications
 open import foundation.dependent-pair-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import foundation-core.function-types
diff --git a/src/foundation/equality-fibers-of-maps.lagda.md b/src/foundation/equality-fibers-of-maps.lagda.md
index aed80bb948..1e45c6ebde 100644
--- a/src/foundation/equality-fibers-of-maps.lagda.md
+++ b/src/foundation/equality-fibers-of-maps.lagda.md
@@ -10,7 +10,7 @@ module foundation.equality-fibers-of-maps where
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import foundation-core.equality-dependent-pair-types
diff --git a/src/foundation/equivalence-relations.lagda.md b/src/foundation/equivalence-relations.lagda.md
index b6c3391c06..05bf364beb 100644
--- a/src/foundation/equivalence-relations.lagda.md
+++ b/src/foundation/equivalence-relations.lagda.md
@@ -39,7 +39,7 @@ open import foundation-core.identity-types
 open import foundation-core.logical-equivalences
 open import foundation-core.propositions
 open import foundation-core.sets
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/exclusive-disjunction.lagda.md b/src/foundation/exclusive-disjunction.lagda.md
index 059f4e02c9..3e1431893c 100644
--- a/src/foundation/exclusive-disjunction.lagda.md
+++ b/src/foundation/exclusive-disjunction.lagda.md
@@ -33,7 +33,7 @@ open import foundation-core.functoriality-dependent-function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.identity-types
 open import foundation-core.propositions
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 
 open import univalent-combinatorics.2-element-types
 open import univalent-combinatorics.equality-finite-types
diff --git a/src/foundation/fiber-inclusions.lagda.md b/src/foundation/fiber-inclusions.lagda.md
index 82de0e1e41..290012dcd7 100644
--- a/src/foundation/fiber-inclusions.lagda.md
+++ b/src/foundation/fiber-inclusions.lagda.md
@@ -12,7 +12,7 @@ open import foundation.cones-over-cospans
 open import foundation.dependent-pair-types
 open import foundation.faithful-maps
 open import foundation.fibers-of-maps
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/foundation/full-subtypes.lagda.md b/src/foundation/full-subtypes.lagda.md
index 57a9714080..da27a78574 100644
--- a/src/foundation/full-subtypes.lagda.md
+++ b/src/foundation/full-subtypes.lagda.md
@@ -16,7 +16,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/function-types.lagda.md b/src/foundation/function-types.lagda.md
index cbafd7364a..a14e912eb0 100644
--- a/src/foundation/function-types.lagda.md
+++ b/src/foundation/function-types.lagda.md
@@ -16,7 +16,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/functoriality-coproduct-types.lagda.md b/src/foundation/functoriality-coproduct-types.lagda.md
index bb0ed02ec6..e534ac71ee 100644
--- a/src/foundation/functoriality-coproduct-types.lagda.md
+++ b/src/foundation/functoriality-coproduct-types.lagda.md
@@ -35,7 +35,7 @@ open import foundation-core.identity-types
 open import foundation-core.injective-maps
 open import foundation-core.negation
 open import foundation-core.propositions
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/functoriality-dependent-function-types.lagda.md b/src/foundation/functoriality-dependent-function-types.lagda.md
index d627e7290b..7d0fb34e88 100644
--- a/src/foundation/functoriality-dependent-function-types.lagda.md
+++ b/src/foundation/functoriality-dependent-function-types.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalence-extensionality
 open import foundation.equivalences
 open import foundation.function-extensionality
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/foundation/functoriality-dependent-pair-types.lagda.md b/src/foundation/functoriality-dependent-pair-types.lagda.md
index 58256ba38a..c5896ad8ed 100644
--- a/src/foundation/functoriality-dependent-pair-types.lagda.md
+++ b/src/foundation/functoriality-dependent-pair-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.pullbacks
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/homotopies.lagda.md b/src/foundation/homotopies.lagda.md
index c29c48ddd9..e42c0af260 100644
--- a/src/foundation/homotopies.lagda.md
+++ b/src/foundation/homotopies.lagda.md
@@ -24,7 +24,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.sections
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/identity-systems.lagda.md b/src/foundation/identity-systems.lagda.md
index 494a0cc6f6..da6502913b 100644
--- a/src/foundation/identity-systems.lagda.md
+++ b/src/foundation/identity-systems.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.contractible-types
 open import foundation-core.equivalences
 open import foundation-core.identity-types
 open import foundation-core.sections
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/identity-types.lagda.md b/src/foundation/identity-types.lagda.md
index 57e77a4bb7..138e0d47c2 100644
--- a/src/foundation/identity-types.lagda.md
+++ b/src/foundation/identity-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
@@ -55,8 +55,8 @@ identifications in arbitrary types.
 | Path algebra                                      | [`foundation.path-algebra`](foundation.path-algebra.md)                                                                   |
 | Symmetric identity types                          | [`foundation.symmetric-identity-types`](foundation.symmetric-identity-types.md)                                           |
 | Torsorial type families                           | [`foundation.torsorial-type-families`](foundation.torsorial-type-families.md)                                             |
-| Transport along identifications (foundation)      | [`foundation.transport-along-identifications`](foundation.transport-along-identifications.md)                             |
-| Transport along identifications (foundation-core) | [`foundation-core.transport-along-identifications`](foundation-core.transport-along-identifications.md)                   |
+| Transport along identifications (foundation)      | [`foundation.transport`](foundation.transport.md)                                                                         |
+| Transport along identifications (foundation-core) | [`foundation-core.transport`](foundation-core.transport.md)                                                               |
 | The universal property of identity types          | [`foundation.universal-property-identity-types`](foundation.universal-property-identity-types.md)                         |
 
 ## Properties
diff --git a/src/foundation/induction-principle-propositional-truncation.lagda.md b/src/foundation/induction-principle-propositional-truncation.lagda.md
index dafe752efa..803ab7d6a0 100644
--- a/src/foundation/induction-principle-propositional-truncation.lagda.md
+++ b/src/foundation/induction-principle-propositional-truncation.lagda.md
@@ -12,7 +12,7 @@ open import foundation.universe-levels
 
 open import foundation-core.identity-types
 open import foundation-core.propositions
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/isolated-points.lagda.md b/src/foundation/isolated-points.lagda.md
index e9196a4deb..0d12b9518e 100644
--- a/src/foundation/isolated-points.lagda.md
+++ b/src/foundation/isolated-points.lagda.md
@@ -35,7 +35,7 @@ open import foundation-core.injective-maps
 open import foundation-core.propositions
 open import foundation-core.sets
 open import foundation-core.subtypes
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/locally-small-types.lagda.md b/src/foundation/locally-small-types.lagda.md
index f16321383d..dfcfb2b8d9 100644
--- a/src/foundation/locally-small-types.lagda.md
+++ b/src/foundation/locally-small-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.small-types
 open import foundation-core.subtypes
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation/partitions.lagda.md b/src/foundation/partitions.lagda.md
index b86e92a187..fa2671d2c3 100644
--- a/src/foundation/partitions.lagda.md
+++ b/src/foundation/partitions.lagda.md
@@ -25,7 +25,7 @@ open import foundation.small-types
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
 open import foundation.surjective-maps
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/foundation/perfect-images.lagda.md b/src/foundation/perfect-images.lagda.md
index d5e2a9986e..586955f0c0 100644
--- a/src/foundation/perfect-images.lagda.md
+++ b/src/foundation/perfect-images.lagda.md
@@ -28,7 +28,7 @@ open import foundation-core.identity-types
 open import foundation-core.injective-maps
 open import foundation-core.propositional-maps
 open import foundation-core.propositions
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/pointed-torsorial-type-families.lagda.md b/src/foundation/pointed-torsorial-type-families.lagda.md
index d8fcdc590e..968bfbf1ca 100644
--- a/src/foundation/pointed-torsorial-type-families.lagda.md
+++ b/src/foundation/pointed-torsorial-type-families.lagda.md
@@ -14,7 +14,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.locally-small-types
 open import foundation.propositional-truncations
 open import foundation.sorial-type-families
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.universe-levels
 
diff --git a/src/foundation/products-unordered-pairs-of-types.lagda.md b/src/foundation/products-unordered-pairs-of-types.lagda.md
index 5741e3ab94..01b4238bfb 100644
--- a/src/foundation/products-unordered-pairs-of-types.lagda.md
+++ b/src/foundation/products-unordered-pairs-of-types.lagda.md
@@ -11,7 +11,7 @@ open import foundation.dependent-pair-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-dependent-function-types
 open import foundation.symmetric-operations
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 open import foundation.unordered-pairs
 open import foundation.unordered-pairs-of-types
diff --git a/src/foundation/propositional-extensionality.lagda.md b/src/foundation/propositional-extensionality.lagda.md
index 0154bd5709..f1c2fd96b1 100644
--- a/src/foundation/propositional-extensionality.lagda.md
+++ b/src/foundation/propositional-extensionality.lagda.md
@@ -15,7 +15,7 @@ open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.raising-universe-levels
 open import foundation.subtype-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md
index a202424801..8342afce15 100644
--- a/src/foundation/propositional-truncations.lagda.md
+++ b/src/foundation/propositional-truncations.lagda.md
@@ -23,7 +23,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.sets
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation/pullbacks.lagda.md b/src/foundation/pullbacks.lagda.md
index 6d4cec21c1..ace83663d3 100644
--- a/src/foundation/pullbacks.lagda.md
+++ b/src/foundation/pullbacks.lagda.md
@@ -31,7 +31,7 @@ open import foundation-core.equality-dependent-pair-types
 open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-function-types
 open import foundation-core.propositions
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/singleton-induction.lagda.md b/src/foundation/singleton-induction.lagda.md
index ed4b0a5879..79b42d767f 100644
--- a/src/foundation/singleton-induction.lagda.md
+++ b/src/foundation/singleton-induction.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.sections
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/structure.lagda.md b/src/foundation/structure.lagda.md
index dea3740006..605062f4e3 100644
--- a/src/foundation/structure.lagda.md
+++ b/src/foundation/structure.lagda.md
@@ -14,7 +14,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.fibers-of-maps
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/subuniverses.lagda.md b/src/foundation/subuniverses.lagda.md
index 5c1d434904..97f5866997 100644
--- a/src/foundation/subuniverses.lagda.md
+++ b/src/foundation/subuniverses.lagda.md
@@ -20,7 +20,7 @@ open import foundation-core.equivalences
 open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/symmetric-difference.lagda.md b/src/foundation/symmetric-difference.lagda.md
index bf91677b74..80f7687779 100644
--- a/src/foundation/symmetric-difference.lagda.md
+++ b/src/foundation/symmetric-difference.lagda.md
@@ -21,7 +21,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index c72dffff47..5a99cda03c 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -14,7 +14,7 @@ open import foundation.equivalence-extensionality
 open import foundation.equivalence-induction
 open import foundation.equivalences
 open import foundation.function-extensionality
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.univalence
 open import foundation.universe-levels
 
@@ -27,9 +27,8 @@ open import foundation-core.identity-types
 
 ## Idea
 
-Applying
-[transport along identifications](foundation-core.transport-along-identifications.md)
-to [identifications](foundation-core.identity-types.md) arising from the
+Applying [transport along identifications](foundation-core.transport.md) to
+[identifications](foundation-core.identity-types.md) arising from the
 [univalence axiom](foundation.univalence.md) gives us **transport along
 equivalences**.
 
diff --git a/src/foundation/transport-along-identifications.lagda.md b/src/foundation/transport.lagda.md
similarity index 94%
rename from src/foundation/transport-along-identifications.lagda.md
rename to src/foundation/transport.lagda.md
index d343c48e48..7c368e818e 100644
--- a/src/foundation/transport-along-identifications.lagda.md
+++ b/src/foundation/transport.lagda.md
@@ -1,9 +1,9 @@
 # Transport along identifications
 
 ```agda
-module foundation.transport-along-identifications where
+module foundation.transport where
 
-open import foundation-core.transport-along-identifications public
+open import foundation-core.transport public
 ```
 
 <details><summary>Imports</summary>
@@ -27,9 +27,8 @@ open import foundation-core.identity-types
 
 Given a type family `B` over `A`, an
 [identification](foundation-core.identity-types.md) `p : x ＝ y` in `A` and an
-element `b : B x`, we can
-[**transport**](foundation-core.transport-along-identifications.md) the element
-`b` along the identification `p` to obtain an element `tr B p b : B y`.
+element `b : B x`, we can [**transport**](foundation-core.transport.md) the
+element `b` along the identification `p` to obtain an element `tr B p b : B y`.
 
 The fact that `tr B p` is an [equivalence](foundation-core.equivalences.md) is
 recorded in this file.
diff --git a/src/foundation/type-theoretic-principle-of-choice.lagda.md b/src/foundation/type-theoretic-principle-of-choice.lagda.md
index 65edfc98a1..1a1a7ef294 100644
--- a/src/foundation/type-theoretic-principle-of-choice.lagda.md
+++ b/src/foundation/type-theoretic-principle-of-choice.lagda.md
@@ -17,7 +17,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/univalence-implies-function-extensionality.lagda.md b/src/foundation/univalence-implies-function-extensionality.lagda.md
index b374530cb1..87a27e5e83 100644
--- a/src/foundation/univalence-implies-function-extensionality.lagda.md
+++ b/src/foundation/univalence-implies-function-extensionality.lagda.md
@@ -20,7 +20,7 @@ open import foundation-core.function-extensionality
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport-along-identifications
+open import foundation-core.transport
 ```
 
 </details>
diff --git a/src/foundation/univalent-type-families.lagda.md b/src/foundation/univalent-type-families.lagda.md
index 8297c1c7a7..5d2e7960cc 100644
--- a/src/foundation/univalent-type-families.lagda.md
+++ b/src/foundation/univalent-type-families.lagda.md
@@ -7,7 +7,7 @@ module foundation.univalent-type-families where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import foundation-core.equivalences
diff --git a/src/foundation/universal-property-image.lagda.md b/src/foundation/universal-property-image.lagda.md
index 1dc5d76fd3..3754713693 100644
--- a/src/foundation/universal-property-image.lagda.md
+++ b/src/foundation/universal-property-image.lagda.md
@@ -13,7 +13,7 @@ open import foundation.images
 open import foundation.propositional-truncations
 open import foundation.slice
 open import foundation.surjective-maps
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import foundation-core.contractible-maps
diff --git a/src/foundation/universal-property-set-quotients.lagda.md b/src/foundation/universal-property-set-quotients.lagda.md
index 1ff24df007..203fa60492 100644
--- a/src/foundation/universal-property-set-quotients.lagda.md
+++ b/src/foundation/universal-property-set-quotients.lagda.md
@@ -25,7 +25,7 @@ open import foundation.propositional-truncations
 open import foundation.reflecting-maps-equivalence-relations
 open import foundation.sets
 open import foundation.surjective-maps
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universal-property-image
 open import foundation.universe-levels
 
diff --git a/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md b/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md
index 74009d5e07..309e845787 100644
--- a/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md
+++ b/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md
@@ -17,7 +17,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import graph-theory.enriched-undirected-graphs
diff --git a/src/graph-theory/equivalences-undirected-graphs.lagda.md b/src/graph-theory/equivalences-undirected-graphs.lagda.md
index 958f6e24b6..050660708e 100644
--- a/src/graph-theory/equivalences-undirected-graphs.lagda.md
+++ b/src/graph-theory/equivalences-undirected-graphs.lagda.md
@@ -20,7 +20,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.univalence
 open import foundation.universe-levels
 open import foundation.unordered-pairs
diff --git a/src/graph-theory/fibers-directed-graphs.lagda.md b/src/graph-theory/fibers-directed-graphs.lagda.md
index 3b7917dec1..2325a19107 100644
--- a/src/graph-theory/fibers-directed-graphs.lagda.md
+++ b/src/graph-theory/fibers-directed-graphs.lagda.md
@@ -14,7 +14,7 @@ open import foundation.equality-dependent-pair-types
 open import foundation.equivalences
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/graph-theory/morphisms-undirected-graphs.lagda.md b/src/graph-theory/morphisms-undirected-graphs.lagda.md
index 004e717c9b..de1a35bfe2 100644
--- a/src/graph-theory/morphisms-undirected-graphs.lagda.md
+++ b/src/graph-theory/morphisms-undirected-graphs.lagda.md
@@ -19,7 +19,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 open import foundation.unordered-pairs
 
diff --git a/src/graph-theory/undirected-graphs.lagda.md b/src/graph-theory/undirected-graphs.lagda.md
index f7b5cbd8f5..cab82a2c23 100644
--- a/src/graph-theory/undirected-graphs.lagda.md
+++ b/src/graph-theory/undirected-graphs.lagda.md
@@ -10,7 +10,7 @@ module graph-theory.undirected-graphs where
 open import foundation.cartesian-product-types
 open import foundation.dependent-pair-types
 open import foundation.equivalences
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 open import foundation.unordered-pairs
 
diff --git a/src/graph-theory/walks-directed-graphs.lagda.md b/src/graph-theory/walks-directed-graphs.lagda.md
index 0b38186d0a..cb327fdab1 100644
--- a/src/graph-theory/walks-directed-graphs.lagda.md
+++ b/src/graph-theory/walks-directed-graphs.lagda.md
@@ -24,7 +24,7 @@ open import foundation.identity-types
 open import foundation.injective-maps
 open import foundation.negation
 open import foundation.raising-universe-levels
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import graph-theory.directed-graphs
diff --git a/src/group-theory/concrete-group-actions.lagda.md b/src/group-theory/concrete-group-actions.lagda.md
index fe407ed17e..bc5108da26 100644
--- a/src/group-theory/concrete-group-actions.lagda.md
+++ b/src/group-theory/concrete-group-actions.lagda.md
@@ -9,7 +9,7 @@ module group-theory.concrete-group-actions where
 ```agda
 open import foundation.function-types
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import group-theory.concrete-groups
diff --git a/src/group-theory/homomorphisms-concrete-group-actions.lagda.md b/src/group-theory/homomorphisms-concrete-group-actions.lagda.md
index 62057b8e87..b86eae7b09 100644
--- a/src/group-theory/homomorphisms-concrete-group-actions.lagda.md
+++ b/src/group-theory/homomorphisms-concrete-group-actions.lagda.md
@@ -16,7 +16,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import group-theory.concrete-group-actions
diff --git a/src/group-theory/normal-subgroups.lagda.md b/src/group-theory/normal-subgroups.lagda.md
index eadcaf09fb..36448d9058 100644
--- a/src/group-theory/normal-subgroups.lagda.md
+++ b/src/group-theory/normal-subgroups.lagda.md
@@ -20,7 +20,7 @@ open import foundation.large-binary-relations
 open import foundation.propositions
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import group-theory.congruence-relations-groups
diff --git a/src/group-theory/subgroups.lagda.md b/src/group-theory/subgroups.lagda.md
index beed392e57..a14391c6d1 100644
--- a/src/group-theory/subgroups.lagda.md
+++ b/src/group-theory/subgroups.lagda.md
@@ -25,7 +25,7 @@ open import foundation.propositions
 open import foundation.sets
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import group-theory.groups
diff --git a/src/group-theory/torsors.lagda.md b/src/group-theory/torsors.lagda.md
index 4a2f87f322..5759ad386f 100644
--- a/src/group-theory/torsors.lagda.md
+++ b/src/group-theory/torsors.lagda.md
@@ -21,7 +21,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtype-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import group-theory.concrete-groups
diff --git a/src/group-theory/transitive-concrete-group-actions.lagda.md b/src/group-theory/transitive-concrete-group-actions.lagda.md
index 2d589c0391..dd3eac590f 100644
--- a/src/group-theory/transitive-concrete-group-actions.lagda.md
+++ b/src/group-theory/transitive-concrete-group-actions.lagda.md
@@ -21,7 +21,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import group-theory.concrete-group-actions
diff --git a/src/higher-group-theory/higher-group-actions.lagda.md b/src/higher-group-theory/higher-group-actions.lagda.md
index 92a3590986..3e339c1a9f 100644
--- a/src/higher-group-theory/higher-group-actions.lagda.md
+++ b/src/higher-group-theory/higher-group-actions.lagda.md
@@ -8,7 +8,7 @@ module higher-group-theory.higher-group-actions where
 
 ```agda
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import higher-group-theory.higher-groups
diff --git a/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md b/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md
index 5929ab0dbc..6d2ee91be6 100644
--- a/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md
+++ b/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md
@@ -12,7 +12,7 @@ open import foundation.equivalences
 open import foundation.function-extensionality
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import higher-group-theory.higher-group-actions
diff --git a/src/linear-algebra/vectors.lagda.md b/src/linear-algebra/vectors.lagda.md
index 4547743267..2fb22bb53a 100644
--- a/src/linear-algebra/vectors.lagda.md
+++ b/src/linear-algebra/vectors.lagda.md
@@ -21,7 +21,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.unit-type
diff --git a/src/lists/functoriality-lists.lagda.md b/src/lists/functoriality-lists.lagda.md
index fe70813734..62b9b8f9c9 100644
--- a/src/lists/functoriality-lists.lagda.md
+++ b/src/lists/functoriality-lists.lagda.md
@@ -16,7 +16,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import linear-algebra.functoriality-vectors
diff --git a/src/lists/permutation-lists.lagda.md b/src/lists/permutation-lists.lagda.md
index b76eba9548..cab237461e 100644
--- a/src/lists/permutation-lists.lagda.md
+++ b/src/lists/permutation-lists.lagda.md
@@ -16,7 +16,7 @@ open import foundation.dependent-pair-types
 open import foundation.equality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import linear-algebra.functoriality-vectors
diff --git a/src/lists/permutation-vectors.lagda.md b/src/lists/permutation-vectors.lagda.md
index ae8d72e68a..16572742fd 100644
--- a/src/lists/permutation-vectors.lagda.md
+++ b/src/lists/permutation-vectors.lagda.md
@@ -22,7 +22,7 @@ open import foundation.function-types
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import linear-algebra.functoriality-vectors
diff --git a/src/lists/sort-by-insertion-vectors.lagda.md b/src/lists/sort-by-insertion-vectors.lagda.md
index 6f756f9386..2f9c6c74e3 100644
--- a/src/lists/sort-by-insertion-vectors.lagda.md
+++ b/src/lists/sort-by-insertion-vectors.lagda.md
@@ -18,7 +18,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/order-theory/finitely-graded-posets.lagda.md b/src/order-theory/finitely-graded-posets.lagda.md
index 7534d942df..2b684e743a 100644
--- a/src/order-theory/finitely-graded-posets.lagda.md
+++ b/src/order-theory/finitely-graded-posets.lagda.md
@@ -24,7 +24,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/organic-chemistry/ethane.lagda.md b/src/organic-chemistry/ethane.lagda.md
index a284da7eff..74afce7451 100644
--- a/src/organic-chemistry/ethane.lagda.md
+++ b/src/organic-chemistry/ethane.lagda.md
@@ -23,7 +23,7 @@ open import foundation.identity-types
 open import foundation.injective-maps
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.univalence
 open import foundation.universe-levels
 open import foundation.unordered-pairs
diff --git a/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md b/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md
index b00fa7a696..c5e9c1501e 100644
--- a/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md
+++ b/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md
@@ -26,7 +26,7 @@ open import foundation.monomorphisms
 open import foundation.propositions
 open import foundation.sets
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-arithmetic-dependent-pair-types
diff --git a/src/orthogonal-factorization-systems/function-classes.lagda.md b/src/orthogonal-factorization-systems/function-classes.lagda.md
index 9c30c83a67..56e789dbf9 100644
--- a/src/orthogonal-factorization-systems/function-classes.lagda.md
+++ b/src/orthogonal-factorization-systems/function-classes.lagda.md
@@ -14,7 +14,7 @@ open import foundation.function-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.pullback-squares
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.univalence
 open import foundation.universe-levels
 ```
diff --git a/src/orthogonal-factorization-systems/open-modalities.lagda.md b/src/orthogonal-factorization-systems/open-modalities.lagda.md
index e6dd80b282..3f0fe930f5 100644
--- a/src/orthogonal-factorization-systems/open-modalities.lagda.md
+++ b/src/orthogonal-factorization-systems/open-modalities.lagda.md
@@ -14,7 +14,7 @@ open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.locally-small-types
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import orthogonal-factorization-systems.higher-modalities
diff --git a/src/polytopes/abstract-polytopes.lagda.md b/src/polytopes/abstract-polytopes.lagda.md
index a490cdeffd..02b69a6446 100644
--- a/src/polytopes/abstract-polytopes.lagda.md
+++ b/src/polytopes/abstract-polytopes.lagda.md
@@ -17,7 +17,7 @@ open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md
index c458e6b9a8..e2f8fdf80b 100644
--- a/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md
+++ b/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md
@@ -17,7 +17,7 @@ open import foundation.logical-equivalences
 open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unions-subtypes
 open import foundation.universe-levels
 
diff --git a/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md
index 3e3e087bfe..a10777ae7b 100644
--- a/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md
+++ b/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md
@@ -18,7 +18,7 @@ open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unions-subtypes
 open import foundation.universe-levels
 
diff --git a/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md
index cec34c1f4e..8e268bd112 100644
--- a/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md
+++ b/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md
@@ -18,7 +18,7 @@ open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unions-subtypes
 open import foundation.universe-levels
 
diff --git a/src/set-theory/cumulative-hierarchy.lagda.md b/src/set-theory/cumulative-hierarchy.lagda.md
index 6ce0932a89..a499a76770 100644
--- a/src/set-theory/cumulative-hierarchy.lagda.md
+++ b/src/set-theory/cumulative-hierarchy.lagda.md
@@ -27,7 +27,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.truncated-types
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md
index 1611c57132..ee599d6e21 100644
--- a/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md
@@ -22,7 +22,7 @@ open import foundation.relaxed-sigma-decompositions
 open import foundation.sigma-closed-subuniverses
 open import foundation.sigma-decomposition-subuniverse
 open import foundation.subuniverses
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-theoretic-principle-of-choice
diff --git a/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md
index 4479561277..050d359dd3 100644
--- a/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md
@@ -22,7 +22,7 @@ open import foundation.relaxed-sigma-decompositions
 open import foundation.sigma-closed-subuniverses
 open import foundation.sigma-decomposition-subuniverse
 open import foundation.subuniverses
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md
index 8a891eacda..50ee887c03 100644
--- a/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md
@@ -20,7 +20,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
diff --git a/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md b/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md
index 025f737803..e7b28ec46c 100644
--- a/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md
@@ -20,7 +20,7 @@ open import foundation.pi-decompositions-subuniverse
 open import foundation.product-decompositions
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.univalence
 open import foundation.universe-levels
diff --git a/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md b/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md
index 2518d8c0bc..4fedc029c4 100644
--- a/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md
@@ -19,7 +19,7 @@ open import foundation.product-decompositions
 open import foundation.product-decompositions-subuniverse
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
diff --git a/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md b/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md
index ff5a2545b2..16a59ec9f6 100644
--- a/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md
@@ -22,7 +22,7 @@ open import foundation.sigma-closed-subuniverses
 open import foundation.sigma-decomposition-subuniverse
 open import foundation.small-types
 open import foundation.subuniverses
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-theoretic-principle-of-choice
diff --git a/src/species/species-of-types-in-subuniverses.lagda.md b/src/species/species-of-types-in-subuniverses.lagda.md
index 1ef1c85fec..890975d154 100644
--- a/src/species/species-of-types-in-subuniverses.lagda.md
+++ b/src/species/species-of-types-in-subuniverses.lagda.md
@@ -13,7 +13,7 @@ open import foundation.equivalences
 open import foundation.function-types
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/species/species-of-types.lagda.md b/src/species/species-of-types.lagda.md
index ceb687541f..585fc15a8c 100644
--- a/src/species/species-of-types.lagda.md
+++ b/src/species/species-of-types.lagda.md
@@ -9,7 +9,7 @@ module species.species-of-types where
 ```agda
 open import foundation.cartesian-product-types
 open import foundation.equivalences
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.univalence
 open import foundation.universe-levels
 ```
diff --git a/src/structured-types/pointed-maps.lagda.md b/src/structured-types/pointed-maps.lagda.md
index ff004374cb..0cdb448f97 100644
--- a/src/structured-types/pointed-maps.lagda.md
+++ b/src/structured-types/pointed-maps.lagda.md
@@ -13,7 +13,7 @@ open import foundation.constant-maps
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import structured-types.pointed-dependent-functions
diff --git a/src/synthetic-homotopy-theory/26-descent.lagda.md b/src/synthetic-homotopy-theory/26-descent.lagda.md
index 713d873a48..1535e6ce6c 100644
--- a/src/synthetic-homotopy-theory/26-descent.lagda.md
+++ b/src/synthetic-homotopy-theory/26-descent.lagda.md
@@ -30,7 +30,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.pullbacks
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.univalence
 open import foundation.universe-levels
diff --git a/src/synthetic-homotopy-theory/26-id-pushout.lagda.md b/src/synthetic-homotopy-theory/26-id-pushout.lagda.md
index 4163a33a94..955cbf3c7b 100644
--- a/src/synthetic-homotopy-theory/26-id-pushout.lagda.md
+++ b/src/synthetic-homotopy-theory/26-id-pushout.lagda.md
@@ -22,7 +22,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universal-property-identity-types
 open import foundation.universe-levels
 
diff --git a/src/synthetic-homotopy-theory/circle.lagda.md b/src/synthetic-homotopy-theory/circle.lagda.md
index 71067c794a..b33a747e72 100644
--- a/src/synthetic-homotopy-theory/circle.lagda.md
+++ b/src/synthetic-homotopy-theory/circle.lagda.md
@@ -16,7 +16,7 @@ open import foundation.identity-types
 open import foundation.mere-equality
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import structured-types.pointed-types
diff --git a/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md b/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md
index fd04ca5e1f..1c6f3f5ea4 100644
--- a/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md
@@ -19,7 +19,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.cocones-under-spans
diff --git a/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md b/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md
index 2017650ebe..96775f12df 100644
--- a/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md
@@ -12,7 +12,7 @@ open import foundation.cones-over-cospans
 open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.pullbacks
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.cocones-under-spans
diff --git a/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md b/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md
index 33e7455434..a3ab28bd9a 100644
--- a/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md b/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md
index b4e7a6a56e..df4f68177f 100644
--- a/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md
@@ -20,7 +20,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.pullbacks
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.cocones-under-spans
diff --git a/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md b/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md
index 0cf2e7b48b..8c6aa3c27e 100644
--- a/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md b/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md
index 4a56bcd36b..15f9bbe907 100644
--- a/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md
+++ b/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md
@@ -17,7 +17,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.functoriality-function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.descent-circle
diff --git a/src/synthetic-homotopy-theory/descent-circle.lagda.md b/src/synthetic-homotopy-theory/descent-circle.lagda.md
index 0fd8cb8fbd..050124368f 100644
--- a/src/synthetic-homotopy-theory/descent-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/descent-circle.lagda.md
@@ -25,7 +25,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.univalence
 open import foundation.universe-levels
@@ -121,8 +121,8 @@ module _
 ### Canonical descent data for a family over the circle
 
 A type family over the circle gives rise to its canonical descent data, obtained
-by evaluation at `base` and
-[transporting](foundation-core.transport-along-identifications.md) along `loop`.
+by evaluation at `base` and [transporting](foundation-core.transport.md) along
+`loop`.
 
 ```agda
 ev-descent-data-circle :
diff --git a/src/synthetic-homotopy-theory/free-loops.lagda.md b/src/synthetic-homotopy-theory/free-loops.lagda.md
index 9fe6f480e3..5e83bb9173 100644
--- a/src/synthetic-homotopy-theory/free-loops.lagda.md
+++ b/src/synthetic-homotopy-theory/free-loops.lagda.md
@@ -16,7 +16,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 ```
 
diff --git a/src/synthetic-homotopy-theory/interval-type.lagda.md b/src/synthetic-homotopy-theory/interval-type.lagda.md
index 6174817856..5013094d8e 100644
--- a/src/synthetic-homotopy-theory/interval-type.lagda.md
+++ b/src/synthetic-homotopy-theory/interval-type.lagda.md
@@ -15,7 +15,7 @@ open import foundation.equivalences
 open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 ```
 
diff --git a/src/synthetic-homotopy-theory/multiplication-circle.lagda.md b/src/synthetic-homotopy-theory/multiplication-circle.lagda.md
index 22827748bd..4dbcf962ac 100644
--- a/src/synthetic-homotopy-theory/multiplication-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/multiplication-circle.lagda.md
@@ -13,7 +13,7 @@ open import foundation.function-extensionality
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import structured-types.pointed-homotopies
diff --git a/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md b/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md
index d1c720d06b..af2782990c 100644
--- a/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md
@@ -23,7 +23,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.path-algebra
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.descent-circle
diff --git a/src/synthetic-homotopy-theory/suspension-structures.lagda.md b/src/synthetic-homotopy-theory/suspension-structures.lagda.md
index 9a931276e9..4e3417d2a0 100644
--- a/src/synthetic-homotopy-theory/suspension-structures.lagda.md
+++ b/src/synthetic-homotopy-theory/suspension-structures.lagda.md
@@ -24,7 +24,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md b/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md
index 66d7735858..af23cf6315 100644
--- a/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md
+++ b/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md b/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md
index b59b336c7c..ec2edeeee3 100644
--- a/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md
+++ b/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md
index 1522eaf184..e9a04aeb4e 100644
--- a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md
@@ -29,7 +29,7 @@ open import foundation.identity-types
 open import foundation.raising-universe-levels
 open import foundation.sets
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.universe-levels
diff --git a/src/synthetic-homotopy-theory/universal-property-circle.lagda.md b/src/synthetic-homotopy-theory/universal-property-circle.lagda.md
index 01e13c3d10..eaeaf94f8f 100644
--- a/src/synthetic-homotopy-theory/universal-property-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-property-circle.lagda.md
@@ -22,7 +22,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sections
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.free-loops
diff --git a/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md b/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
index c3ca4487dd..a608a985a0 100644
--- a/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md b/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md
index 4d9bbdad38..3cd5f42732 100644
--- a/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/trees/bases-directed-trees.lagda.md b/src/trees/bases-directed-trees.lagda.md
index 62493b1ade..fa49199117 100644
--- a/src/trees/bases-directed-trees.lagda.md
+++ b/src/trees/bases-directed-trees.lagda.md
@@ -21,7 +21,7 @@ open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
 open import foundation.universe-levels
diff --git a/src/trees/directed-trees.lagda.md b/src/trees/directed-trees.lagda.md
index 629fb4c5bb..8c72d592d4 100644
--- a/src/trees/directed-trees.lagda.md
+++ b/src/trees/directed-trees.lagda.md
@@ -25,7 +25,7 @@ open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
diff --git a/src/trees/elementhood-relation-w-types.lagda.md b/src/trees/elementhood-relation-w-types.lagda.md
index 2e980645a5..342b795a11 100644
--- a/src/trees/elementhood-relation-w-types.lagda.md
+++ b/src/trees/elementhood-relation-w-types.lagda.md
@@ -10,7 +10,7 @@ module trees.elementhood-relation-w-types where
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import trees.elementhood-relation-coalgebras-polynomial-endofunctors
diff --git a/src/trees/enriched-directed-trees.lagda.md b/src/trees/enriched-directed-trees.lagda.md
index 570286771d..57fbf3744d 100644
--- a/src/trees/enriched-directed-trees.lagda.md
+++ b/src/trees/enriched-directed-trees.lagda.md
@@ -20,7 +20,7 @@ open import foundation.identity-types
 open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import graph-theory.directed-graphs
diff --git a/src/trees/equivalences-directed-trees.lagda.md b/src/trees/equivalences-directed-trees.lagda.md
index c508da1874..89badad851 100644
--- a/src/trees/equivalences-directed-trees.lagda.md
+++ b/src/trees/equivalences-directed-trees.lagda.md
@@ -19,7 +19,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.subtype-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import graph-theory.equivalences-directed-graphs
diff --git a/src/trees/equivalences-enriched-directed-trees.lagda.md b/src/trees/equivalences-enriched-directed-trees.lagda.md
index c796443d76..e24e0e65d7 100644
--- a/src/trees/equivalences-enriched-directed-trees.lagda.md
+++ b/src/trees/equivalences-enriched-directed-trees.lagda.md
@@ -20,7 +20,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import trees.enriched-directed-trees
diff --git a/src/trees/extensional-w-types.lagda.md b/src/trees/extensional-w-types.lagda.md
index 3a8887dc5d..d53f7b4c1a 100644
--- a/src/trees/extensional-w-types.lagda.md
+++ b/src/trees/extensional-w-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.slice
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.univalent-type-families
 open import foundation.universe-levels
diff --git a/src/trees/functoriality-combinator-directed-trees.lagda.md b/src/trees/functoriality-combinator-directed-trees.lagda.md
index 62ecdf2b9c..b181907762 100644
--- a/src/trees/functoriality-combinator-directed-trees.lagda.md
+++ b/src/trees/functoriality-combinator-directed-trees.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import trees.combinator-directed-trees
diff --git a/src/trees/functoriality-w-types.lagda.md b/src/trees/functoriality-w-types.lagda.md
index 51e33d255b..f55e1f0957 100644
--- a/src/trees/functoriality-w-types.lagda.md
+++ b/src/trees/functoriality-w-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.functoriality-dependent-function-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositional-maps
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.truncated-maps
 open import foundation.truncated-types
 open import foundation.truncation-levels
diff --git a/src/trees/induction-w-types.lagda.md b/src/trees/induction-w-types.lagda.md
index 1805b55311..bf924b6e13 100644
--- a/src/trees/induction-w-types.lagda.md
+++ b/src/trees/induction-w-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation.function-extensionality
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import trees.elementhood-relation-w-types
diff --git a/src/trees/morphisms-directed-trees.lagda.md b/src/trees/morphisms-directed-trees.lagda.md
index eea276a027..f4cb1d987a 100644
--- a/src/trees/morphisms-directed-trees.lagda.md
+++ b/src/trees/morphisms-directed-trees.lagda.md
@@ -17,7 +17,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-empty-type
 open import foundation.universe-levels
 
diff --git a/src/trees/morphisms-enriched-directed-trees.lagda.md b/src/trees/morphisms-enriched-directed-trees.lagda.md
index c6618ae770..ea3167e9b1 100644
--- a/src/trees/morphisms-enriched-directed-trees.lagda.md
+++ b/src/trees/morphisms-enriched-directed-trees.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import trees.enriched-directed-trees
diff --git a/src/trees/polynomial-endofunctors.lagda.md b/src/trees/polynomial-endofunctors.lagda.md
index d2d8d3280d..96be64542a 100644
--- a/src/trees/polynomial-endofunctors.lagda.md
+++ b/src/trees/polynomial-endofunctors.lagda.md
@@ -16,7 +16,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 ```
 
diff --git a/src/trees/ranks-of-elements-w-types.lagda.md b/src/trees/ranks-of-elements-w-types.lagda.md
index 1efe2269d1..4ef53cb626 100644
--- a/src/trees/ranks-of-elements-w-types.lagda.md
+++ b/src/trees/ranks-of-elements-w-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import trees.elementhood-relation-w-types
diff --git a/src/trees/rooted-morphisms-directed-trees.lagda.md b/src/trees/rooted-morphisms-directed-trees.lagda.md
index aad2dd69aa..2b46217d61 100644
--- a/src/trees/rooted-morphisms-directed-trees.lagda.md
+++ b/src/trees/rooted-morphisms-directed-trees.lagda.md
@@ -18,7 +18,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subtype-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import trees.bases-directed-trees
diff --git a/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md b/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md
index f876ff518a..1836a44f6d 100644
--- a/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md
+++ b/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md
@@ -12,7 +12,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import trees.enriched-directed-trees
diff --git a/src/trees/small-multisets.lagda.md b/src/trees/small-multisets.lagda.md
index acfd2d4d2c..63341f8813 100644
--- a/src/trees/small-multisets.lagda.md
+++ b/src/trees/small-multisets.lagda.md
@@ -18,7 +18,7 @@ open import foundation.identity-types
 open import foundation.propositions
 open import foundation.small-types
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md b/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md
index 51c5d408dd..5a420b099b 100644
--- a/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md
+++ b/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md
@@ -23,7 +23,7 @@ open import foundation.identity-types
 open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-empty-type
 open import foundation.universe-levels
 
diff --git a/src/trees/underlying-trees-of-elements-of-w-types.lagda.md b/src/trees/underlying-trees-of-elements-of-w-types.lagda.md
index 6579f766cd..2a196afe79 100644
--- a/src/trees/underlying-trees-of-elements-of-w-types.lagda.md
+++ b/src/trees/underlying-trees-of-elements-of-w-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.identity-types
 open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import graph-theory.directed-graphs
diff --git a/src/trees/undirected-trees.lagda.md b/src/trees/undirected-trees.lagda.md
index b9317cfec2..539065d90d 100644
--- a/src/trees/undirected-trees.lagda.md
+++ b/src/trees/undirected-trees.lagda.md
@@ -20,7 +20,7 @@ open import foundation.mere-equality
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import graph-theory.paths-undirected-graphs
diff --git a/src/trees/universal-multiset.lagda.md b/src/trees/universal-multiset.lagda.md
index 274f1dfe83..126ca1365c 100644
--- a/src/trees/universal-multiset.lagda.md
+++ b/src/trees/universal-multiset.lagda.md
@@ -13,7 +13,7 @@ open import foundation.identity-types
 open import foundation.raising-universe-levels
 open import foundation.small-types
 open import foundation.small-universes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import trees.functoriality-w-types
diff --git a/src/trees/w-types.lagda.md b/src/trees/w-types.lagda.md
index e8a3bd5de9..54a11efe7d 100644
--- a/src/trees/w-types.lagda.md
+++ b/src/trees/w-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-theoretic-principle-of-choice
diff --git a/src/type-theories/dependent-type-theories.lagda.md b/src/type-theories/dependent-type-theories.lagda.md
index 3c9f43e8b1..9fb662749e 100644
--- a/src/type-theories/dependent-type-theories.lagda.md
+++ b/src/type-theories/dependent-type-theories.lagda.md
@@ -17,7 +17,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 ```
 
diff --git a/src/type-theories/fibered-dependent-type-theories.lagda.md b/src/type-theories/fibered-dependent-type-theories.lagda.md
index a5da36d53a..fef7eb8f07 100644
--- a/src/type-theories/fibered-dependent-type-theories.lagda.md
+++ b/src/type-theories/fibered-dependent-type-theories.lagda.md
@@ -13,7 +13,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import type-theories.dependent-type-theories
diff --git a/src/type-theories/sections-dependent-type-theories.lagda.md b/src/type-theories/sections-dependent-type-theories.lagda.md
index 339dc4c6e7..57a5c18782 100644
--- a/src/type-theories/sections-dependent-type-theories.lagda.md
+++ b/src/type-theories/sections-dependent-type-theories.lagda.md
@@ -10,7 +10,7 @@ module type-theories.sections-dependent-type-theories where
 
 ```agda
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import type-theories.dependent-type-theories
diff --git a/src/type-theories/simple-type-theories.lagda.md b/src/type-theories/simple-type-theories.lagda.md
index 98461a1a5e..227dc224ff 100644
--- a/src/type-theories/simple-type-theories.lagda.md
+++ b/src/type-theories/simple-type-theories.lagda.md
@@ -13,7 +13,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 ```
 
diff --git a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md
index 335deed2c7..4f925227c3 100644
--- a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md
+++ b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md
@@ -34,7 +34,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.univalence
 open import foundation.universe-levels
diff --git a/src/univalent-combinatorics/2-element-subtypes.lagda.md b/src/univalent-combinatorics/2-element-subtypes.lagda.md
index d41430d2f4..5ba14b8c22 100644
--- a/src/univalent-combinatorics/2-element-subtypes.lagda.md
+++ b/src/univalent-combinatorics/2-element-subtypes.lagda.md
@@ -24,7 +24,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/univalent-combinatorics/2-element-types.lagda.md b/src/univalent-combinatorics/2-element-types.lagda.md
index ed91e496bd..19d458a7af 100644
--- a/src/univalent-combinatorics/2-element-types.lagda.md
+++ b/src/univalent-combinatorics/2-element-types.lagda.md
@@ -37,7 +37,7 @@ open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.sets
 open import foundation.subuniverses
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
diff --git a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md
index 6e970321cd..db7c453f7b 100644
--- a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md
+++ b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md
@@ -26,7 +26,7 @@ open import foundation.functoriality-coproduct-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.sections
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-unit-type
diff --git a/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md b/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md
index a93706de19..f916a53857 100644
--- a/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md
+++ b/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md
@@ -22,7 +22,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.iterated-cartesian-product-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/univalent-combinatorics/decidable-subtypes.lagda.md b/src/univalent-combinatorics/decidable-subtypes.lagda.md
index 66b79ec69f..90a64f18aa 100644
--- a/src/univalent-combinatorics/decidable-subtypes.lagda.md
+++ b/src/univalent-combinatorics/decidable-subtypes.lagda.md
@@ -22,7 +22,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import univalent-combinatorics.decidable-dependent-pair-types
diff --git a/src/univalent-combinatorics/dependent-pair-types.lagda.md b/src/univalent-combinatorics/dependent-pair-types.lagda.md
index 5ab79d007e..d7acd5e852 100644
--- a/src/univalent-combinatorics/dependent-pair-types.lagda.md
+++ b/src/univalent-combinatorics/dependent-pair-types.lagda.md
@@ -25,7 +25,7 @@ open import foundation.propositions
 open import foundation.sections
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
diff --git a/src/univalent-combinatorics/equivalences-cubes.lagda.md b/src/univalent-combinatorics/equivalences-cubes.lagda.md
index f71497fe95..89f968dd7a 100644
--- a/src/univalent-combinatorics/equivalences-cubes.lagda.md
+++ b/src/univalent-combinatorics/equivalences-cubes.lagda.md
@@ -19,7 +19,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.universe-levels
 
 open import univalent-combinatorics.cubes
diff --git a/src/univalent-combinatorics/fibers-of-maps.lagda.md b/src/univalent-combinatorics/fibers-of-maps.lagda.md
index 7ad1cdbb42..79a868aaca 100644
--- a/src/univalent-combinatorics/fibers-of-maps.lagda.md
+++ b/src/univalent-combinatorics/fibers-of-maps.lagda.md
@@ -21,7 +21,7 @@ open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sections
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/univalent-combinatorics/finite-types.lagda.md b/src/univalent-combinatorics/finite-types.lagda.md
index 4fa2bac999..2791564474 100644
--- a/src/univalent-combinatorics/finite-types.lagda.md
+++ b/src/univalent-combinatorics/finite-types.lagda.md
@@ -32,7 +32,7 @@ open import foundation.raising-universe-levels
 open import foundation.sets
 open import foundation.subtypes
 open import foundation.subuniverses
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md
index 9f1ddb69ab..8dc231e357 100644
--- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md
+++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md
@@ -48,7 +48,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/univalent-combinatorics/pi-finite-types.lagda.md b/src/univalent-combinatorics/pi-finite-types.lagda.md
index fb068418ad..d1adf3e29c 100644
--- a/src/univalent-combinatorics/pi-finite-types.lagda.md
+++ b/src/univalent-combinatorics/pi-finite-types.lagda.md
@@ -46,7 +46,7 @@ open import foundation.set-truncations
 open import foundation.sets
 open import foundation.subtypes
 open import foundation.surjective-maps
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-arithmetic-coproduct-types
diff --git a/src/universal-algebra/terms-over-signatures.lagda.md b/src/universal-algebra/terms-over-signatures.lagda.md
index 8a5cf6febf..48f19c048d 100644
--- a/src/universal-algebra/terms-over-signatures.lagda.md
+++ b/src/universal-algebra/terms-over-signatures.lagda.md
@@ -14,7 +14,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport-along-identifications
+open import foundation.transport
 open import foundation.unit-type
 open import foundation.universe-levels
 

From f1aebe76c4dd851383458899bc9bb600e6a1d899 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 11 Sep 2023 16:45:31 +0200
Subject: [PATCH 28/37] resolve comments

---
 ...rtier-delooping-sign-homomorphism.lagda.md |  2 +-
 .../delooping-sign-homomorphism.lagda.md      |  2 +-
 ...mpson-delooping-sign-homomorphism.lagda.md |  2 +-
 src/foundation.lagda.md                       |  2 +-
 ...ences-families-over-subuniverses.lagda.md} |  4 +-
 .../action-on-equivalences-functions.lagda.md | 38 +++++++++----------
 6 files changed, 25 insertions(+), 25 deletions(-)
 rename src/foundation/{action-on-equivalences-subuniverses.lagda.md => action-on-equivalences-families-over-subuniverses.lagda.md} (97%)

diff --git a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
index 5da4a8f958..6ee9081c2d 100644
--- a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
@@ -17,7 +17,7 @@ open import finite-group-theory.finite-type-groups
 open import finite-group-theory.sign-homomorphism
 open import finite-group-theory.transpositions
 
-open import foundation.action-on-equivalences-subuniverses
+open import foundation.action-on-equivalences-families-over-subuniverses
 open import foundation.action-on-identifications-functions
 open import foundation.contractible-types
 open import foundation.dependent-pair-types
diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index a4c900d575..4dd397a5a4 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -18,7 +18,7 @@ open import finite-group-theory.permutations
 open import finite-group-theory.sign-homomorphism
 open import finite-group-theory.transpositions
 
-open import foundation.action-on-equivalences-subuniverses
+open import foundation.action-on-equivalences-families-over-subuniverses
 open import foundation.action-on-identifications-functions
 open import foundation.commuting-squares-of-maps
 open import foundation.contractible-types
diff --git a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
index 54076b09ab..2f36249fd8 100644
--- a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
@@ -21,7 +21,7 @@ open import finite-group-theory.permutations
 open import finite-group-theory.sign-homomorphism
 open import finite-group-theory.transpositions
 
-open import foundation.action-on-equivalences-subuniverses
+open import foundation.action-on-equivalences-families-over-subuniverses
 open import foundation.action-on-identifications-functions
 open import foundation.contractible-types
 open import foundation.coproduct-types
diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index 81c66a1f60..d8517a4065 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -11,7 +11,7 @@ open import foundation.0-maps public
 open import foundation.1-types public
 open import foundation.2-types public
 open import foundation.action-on-equivalences-functions public
-open import foundation.action-on-equivalences-subuniverses public
+open import foundation.action-on-equivalences-families-over-subuniverses public
 open import foundation.action-on-identifications-binary-functions public
 open import foundation.action-on-identifications-dependent-functions public
 open import foundation.action-on-identifications-functions public
diff --git a/src/foundation/action-on-equivalences-subuniverses.lagda.md b/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
similarity index 97%
rename from src/foundation/action-on-equivalences-subuniverses.lagda.md
rename to src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
index b9f13a1e86..fae0017d26 100644
--- a/src/foundation/action-on-equivalences-subuniverses.lagda.md
+++ b/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
@@ -1,7 +1,7 @@
-# Action on equivalences in subuniverses
+# Action on equivalences in type families over subuniverses
 
 ```agda
-module foundation.action-on-equivalences-subuniverses where
+module foundation.action-on-equivalences-families-over-subuniverses where
 ```
 
 <details><summary>Imports</summary>
diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index 9ff204868d..d7dfecf400 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -94,23 +94,37 @@ compute-ap-equiv-id-equiv :
 compute-ap-equiv-id-equiv f A = ap equiv-eq (compute-eq-ap-equiv-id-equiv f A)
 ```
 
+### The action on equivalences of a constant map is constant
+
+```agda
+compute-eq-ap-equiv-const :
+  {l1 l2 : Level} {B : UU l2} (b : B) {X Y : UU l1}
+  (e : X ≃ Y) → (eq-ap-equiv (const (UU l1) B b) e) ＝ refl
+compute-eq-ap-equiv-const b {X} {Y} e = ap-const b (eq-equiv X Y e)
+
+compute-ap-equiv-const :
+  {l1 l2 : Level} (B : UU l2) {X Y : UU l1}
+  (e : X ≃ Y) → (ap-equiv (const (UU l1) (UU l2) B) e) ＝ id-equiv
+compute-ap-equiv-const B {X} {Y} e = ap equiv-eq (compute-eq-ap-equiv-const B e)
+```
+
 ### The action on equivalences of any map preserves composition of equivalences
 
 ```agda
-distributive-eq-ap-equiv-equiv-comp :
+distributive-eq-ap-equiv-comp-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y Z : UU l1} →
   (e : X ≃ Y) (e' : Y ≃ Z) →
   eq-ap-equiv f (e' ∘e e) ＝ (eq-ap-equiv f e ∙ eq-ap-equiv f e')
-distributive-eq-ap-equiv-equiv-comp f {X} {Y} {Z} e e' =
+distributive-eq-ap-equiv-comp-equiv f {X} {Y} {Z} e e' =
     ( ap (ap f) (inv (compute-eq-equiv-equiv-comp X Y Z e e'))) ∙
     ( ap-concat f (eq-equiv X Y e) (eq-equiv Y Z e'))
 
-distributive-ap-equiv-equiv-comp :
+distributive-ap-equiv-comp-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y Z : UU l1} →
   (e : X ≃ Y) (e' : Y ≃ Z) →
   ap-equiv f (e' ∘e e) ＝ (ap-equiv f e' ∘e ap-equiv f e)
-distributive-ap-equiv-equiv-comp f {X} {Y} {Z} e e' =
-  ( ap equiv-eq (distributive-eq-ap-equiv-equiv-comp f e e')) ∙
+distributive-ap-equiv-comp-equiv f {X} {Y} {Z} e e' =
+  ( ap equiv-eq (distributive-eq-ap-equiv-comp-equiv f e e')) ∙
   ( inv (compute-equiv-eq-concat (eq-ap-equiv f e) (eq-ap-equiv f e')))
 ```
 
@@ -131,17 +145,3 @@ compute-ap-equiv-inv-equiv f {X} {Y} e =
   ( ap equiv-eq (compute-eq-ap-equiv-inv-equiv f e)) ∙
   ( inv (commutativity-inv-equiv-eq (f X) (f Y) (eq-ap-equiv f e)))
 ```
-
-### The action on equivalences of a constant map is constant
-
-```agda
-compute-eq-ap-equiv-const :
-  {l1 l2 : Level} {B : UU l2} (b : B) {X Y : UU l1}
-  (e : X ≃ Y) → (eq-ap-equiv (const (UU l1) B b) e) ＝ refl
-compute-eq-ap-equiv-const b {X} {Y} e = ap-const b (eq-equiv X Y e)
-
-compute-ap-equiv-const :
-  {l1 l2 : Level} (B : UU l2) {X Y : UU l1}
-  (e : X ≃ Y) → (ap-equiv (const (UU l1) (UU l2) B) e) ＝ id-equiv
-compute-ap-equiv-const B {X} {Y} e = ap equiv-eq (compute-eq-ap-equiv-const B e)
-```

From f276bc1c4f64cb9adb81f367264122f42be0efda Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 11 Sep 2023 16:45:46 +0200
Subject: [PATCH 29/37] pre-commit

---
 src/foundation.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index d8517a4065..ddec2d8c87 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -10,8 +10,8 @@ open import foundation.0-images-of-maps public
 open import foundation.0-maps public
 open import foundation.1-types public
 open import foundation.2-types public
-open import foundation.action-on-equivalences-functions public
 open import foundation.action-on-equivalences-families-over-subuniverses public
+open import foundation.action-on-equivalences-functions public
 open import foundation.action-on-identifications-binary-functions public
 open import foundation.action-on-identifications-dependent-functions public
 open import foundation.action-on-identifications-functions public

From e6706864e52d0c138037de4d7fbec58470c0ceac Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 11 Sep 2023 16:52:58 +0200
Subject: [PATCH 30/37] revert to `inv-tr`

---
 src/foundation/transport.lagda.md                      | 10 +++++-----
 .../universal-property-set-quotients.lagda.md          |  2 +-
 2 files changed, 6 insertions(+), 6 deletions(-)

diff --git a/src/foundation/transport.lagda.md b/src/foundation/transport.lagda.md
index 4234d1daf1..17957496de 100644
--- a/src/foundation/transport.lagda.md
+++ b/src/foundation/transport.lagda.md
@@ -63,19 +63,19 @@ module _
   {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {x y : A}
   where
 
-  tr-inv : x ＝ y → B y → B x
-  tr-inv p = tr B (inv p)
+  inv-tr : x ＝ y → B y → B x
+  inv-tr p = tr B (inv p)
 
-  is-section-tr : (p : x ＝ y) → (tr-inv p ∘ tr B p) ~ id
+  is-section-tr : (p : x ＝ y) → (inv-tr p ∘ tr B p) ~ id
   is-section-tr refl b = refl
 
-  is-retraction-tr : (p : x ＝ y) → (tr B p ∘ tr-inv p) ~ id
+  is-retraction-tr : (p : x ＝ y) → (tr B p ∘ inv-tr p) ~ id
   is-retraction-tr refl b = refl
 
   is-equiv-tr : (p : x ＝ y) → is-equiv (tr B p)
   is-equiv-tr p =
     is-equiv-is-invertible
-      ( tr-inv p)
+      ( inv-tr p)
       ( is-retraction-tr p)
       ( is-section-tr p)
 
diff --git a/src/foundation/universal-property-set-quotients.lagda.md b/src/foundation/universal-property-set-quotients.lagda.md
index 0f5096ff32..ea589810ab 100644
--- a/src/foundation/universal-property-set-quotients.lagda.md
+++ b/src/foundation/universal-property-set-quotients.lagda.md
@@ -460,7 +460,7 @@ module _
             ( reflects-map-reflecting-map-Equivalence-Relation R q
               ( map-inv-equiv
                 ( compute-P (pr1 v))
-                ( tr-inv (type-Prop ∘ P) (pr2 v) p))) ∙
+                ( inv-tr (type-Prop ∘ P) (pr2 v) p))) ∙
             ( pr2 v)))
     is-contr-total-P : is-contr (Σ (type-Set B) (λ b → type-Prop (P b)))
     is-contr-total-P = pair center-total-P contraction-total-P

From 124c77ccc2b68a52723b1780a605a13c983fb2cf Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 11 Sep 2023 16:55:04 +0200
Subject: [PATCH 31/37] revert to `inv-tr` 2

---
 src/foundation/transport-along-equivalences.lagda.md |  4 ++--
 src/foundation/transport.lagda.md                    | 12 ++++++------
 src/lists/permutation-lists.lagda.md                 |  2 +-
 3 files changed, 9 insertions(+), 9 deletions(-)

diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 5a99cda03c..01e9f0b7c5 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -110,7 +110,7 @@ is-section-map-tr-equiv f {X} {Y} e x =
   ( ap
     ( λ p → tr f p (map-tr-equiv f e x))
     ( inv (commutativity-inv-eq-equiv X Y e))) ∙
-  ( is-section-tr f (eq-equiv X Y e) x)
+  ( is-retraction-inv-tr f (eq-equiv X Y e) x)
 
 is-retraction-map-tr-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2)
@@ -120,7 +120,7 @@ is-retraction-map-tr-equiv f {X} {Y} e x =
   ( ap
     ( map-tr-equiv f e ∘ (λ p → tr f p x))
     ( inv (commutativity-inv-eq-equiv X Y e))) ∙
-  ( is-retraction-tr f (eq-equiv X Y e) x)
+  ( is-section-inv-tr f (eq-equiv X Y e) x)
 ```
 
 ### Transposing transport along the inverse of an equivalence
diff --git a/src/foundation/transport.lagda.md b/src/foundation/transport.lagda.md
index 17957496de..ca68f5dda5 100644
--- a/src/foundation/transport.lagda.md
+++ b/src/foundation/transport.lagda.md
@@ -66,18 +66,18 @@ module _
   inv-tr : x ＝ y → B y → B x
   inv-tr p = tr B (inv p)
 
-  is-section-tr : (p : x ＝ y) → (inv-tr p ∘ tr B p) ~ id
-  is-section-tr refl b = refl
+  is-retraction-inv-tr : (p : x ＝ y) → (inv-tr p ∘ tr B p) ~ id
+  is-retraction-inv-tr refl b = refl
 
-  is-retraction-tr : (p : x ＝ y) → (tr B p ∘ inv-tr p) ~ id
-  is-retraction-tr refl b = refl
+  is-section-inv-tr : (p : x ＝ y) → (tr B p ∘ inv-tr p) ~ id
+  is-section-inv-tr refl b = refl
 
   is-equiv-tr : (p : x ＝ y) → is-equiv (tr B p)
   is-equiv-tr p =
     is-equiv-is-invertible
       ( inv-tr p)
-      ( is-retraction-tr p)
-      ( is-section-tr p)
+      ( is-section-inv-tr p)
+      ( is-retraction-inv-tr p)
 
   equiv-tr : x ＝ y → (B x) ≃ (B y)
   pr1 (equiv-tr p) = tr B p
diff --git a/src/lists/permutation-lists.lagda.md b/src/lists/permutation-lists.lagda.md
index cab237461e..32fb571f8d 100644
--- a/src/lists/permutation-lists.lagda.md
+++ b/src/lists/permutation-lists.lagda.md
@@ -289,7 +289,7 @@ eq-map-list-permute-list {B = B} f p t =
               ( t)) ∙
             ( ap
               ( λ v → permute-vec (length-list p) v t)
-              ( is-retraction-tr
+              ( is-section-inv-tr
                 ( vec B)
                 ( length-map-list f p)
                 ( map-vec f (vec-list p)))))))) ∙

From e678e645ed5188d573035223189d8df989371963 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 11 Sep 2023 16:57:01 +0200
Subject: [PATCH 32/37] `substitution-law-tr`

---
 src/foundation/equality-dependent-pair-types.lagda.md | 2 +-
 src/foundation/transport-along-equivalences.lagda.md  | 6 +++---
 src/foundation/transport.lagda.md                     | 4 ++--
 3 files changed, 6 insertions(+), 6 deletions(-)

diff --git a/src/foundation/equality-dependent-pair-types.lagda.md b/src/foundation/equality-dependent-pair-types.lagda.md
index 0de1851187..ab452b7651 100644
--- a/src/foundation/equality-dependent-pair-types.lagda.md
+++ b/src/foundation/equality-dependent-pair-types.lagda.md
@@ -100,7 +100,7 @@ module _
     {x y : X} (p : x ＝ y) →
     pair-eq-Σ (ap f p) ＝
     ( ( ap (pr1 ∘ f) p) ,
-      ( substitution-tr B (pr1 ∘ f) p ∙ apd (pr2 ∘ f) p))
+      ( substitution-law-tr B (pr1 ∘ f) p ∙ apd (pr2 ∘ f) p))
   pair-eq-Σ-ap refl = refl
 
   pr1-pair-eq-Σ-ap :
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 01e9f0b7c5..587367c1fb 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -149,12 +149,12 @@ substitution-map-tr-equiv :
   (e : X ≃ Y) → map-tr-equiv g (ap-equiv f e) ~ map-tr-equiv (g ∘ f) e
 substitution-map-tr-equiv g f {X} {Y} e X' =
   ( ap (λ p → tr g p X') (is-retraction-eq-equiv (eq-ap-equiv f e))) ∙
-  ( substitution-tr g f (eq-equiv X Y e))
+  ( substitution-law-tr g f (eq-equiv X Y e))
 
-substitution-tr-equiv :
+substitution-law-tr-equiv :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2) {X Y : UU l1}
   (e : X ≃ Y) → tr-equiv g (ap-equiv f e) ＝ tr-equiv (g ∘ f) e
-substitution-tr-equiv g f e = eq-htpy-equiv (substitution-map-tr-equiv g f e)
+substitution-law-tr-equiv g f e = eq-htpy-equiv (substitution-map-tr-equiv g f e)
 ```
 
 ### Transporting along the action on equivalences of a function
diff --git a/src/foundation/transport.lagda.md b/src/foundation/transport.lagda.md
index ca68f5dda5..3d073a2711 100644
--- a/src/foundation/transport.lagda.md
+++ b/src/foundation/transport.lagda.md
@@ -96,11 +96,11 @@ equiv-tr-refl B = refl
 ### Substitution law for transport
 
 ```agda
-substitution-tr :
+substitution-law-tr :
   {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} (B : A → UU l3) (f : X → A)
   {x y : X} (p : x ＝ y) {x' : B (f x)} →
   tr B (ap f p) x' ＝ tr (B ∘ f) p x'
-substitution-tr B f refl = refl
+substitution-law-tr B f refl = refl
 ```
 
 ### Coherences and algebraic identities for `tr²`

From 0f9af0b113042f2cf08e0abbdb55e119a621e950 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 11 Sep 2023 16:59:11 +0200
Subject: [PATCH 33/37] `comp-equiv`

---
 src/finite-group-theory/delooping-sign-homomorphism.lagda.md | 4 ++--
 src/foundation/action-on-equivalences-functions.lagda.md     | 2 +-
 src/foundation/transport-along-equivalences.lagda.md         | 2 +-
 src/foundation/univalence.lagda.md                           | 4 ++--
 src/group-theory/loop-groups-sets.lagda.md                   | 2 +-
 5 files changed, 7 insertions(+), 7 deletions(-)

diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index 4dd397a5a4..5e97d09b16 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -1098,7 +1098,7 @@ module _
             ( λ r → eq-pair-Σ r (eq-is-prop is-prop-type-trunc-Prop))
             ( ap inv
               ( inv
-                ( compute-eq-equiv-equiv-comp
+                ( compute-eq-equiv-comp-equiv
                   ( raise l4 (Fin 2))
                   ( equivalence-class (R (n +ℕ 2) (Fin-UU-Fin l1 (n +ℕ 2))))
                   ( raise l4 (Fin 2))
@@ -1113,7 +1113,7 @@ module _
                 ( ap
                   ( _∙ eq-counting-equivalence-class-R n)
                   ( inv
-                    ( compute-eq-equiv-equiv-comp
+                    ( compute-eq-equiv-comp-equiv
                       ( raise l4 (Fin 2))
                       ( equivalence-class (R (n +ℕ 2) (Fin-UU-Fin l1 (n +ℕ 2))))
                       ( equivalence-class (R (n +ℕ 2) (Fin-UU-Fin l1 (n +ℕ 2))))
diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index d7dfecf400..91a89b5201 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -116,7 +116,7 @@ distributive-eq-ap-equiv-comp-equiv :
   (e : X ≃ Y) (e' : Y ≃ Z) →
   eq-ap-equiv f (e' ∘e e) ＝ (eq-ap-equiv f e ∙ eq-ap-equiv f e')
 distributive-eq-ap-equiv-comp-equiv f {X} {Y} {Z} e e' =
-    ( ap (ap f) (inv (compute-eq-equiv-equiv-comp X Y Z e e'))) ∙
+    ( ap (ap f) (inv (compute-eq-equiv-comp-equiv X Y Z e e'))) ∙
     ( ap-concat f (eq-equiv X Y e) (eq-equiv Y Z e'))
 
 distributive-ap-equiv-comp-equiv :
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 587367c1fb..7c7d603135 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -88,7 +88,7 @@ distributive-map-tr-equiv-equiv-comp :
   {X Y Z : UU l1} (e : X ≃ Y) (e' : Y ≃ Z) →
   map-tr-equiv f (e' ∘e e) ~ (map-tr-equiv f e' ∘ map-tr-equiv f e)
 distributive-map-tr-equiv-equiv-comp f {X} {Y} {Z} e e' x =
-  ( ap (λ p → tr f p x) (inv (compute-eq-equiv-equiv-comp X Y Z e e'))) ∙
+  ( ap (λ p → tr f p x) (inv (compute-eq-equiv-comp-equiv X Y Z e e'))) ∙
   ( tr-concat (eq-equiv X Y e) (eq-equiv Y Z e') x)
 
 distributive-tr-equiv-equiv-comp :
diff --git a/src/foundation/univalence.lagda.md b/src/foundation/univalence.lagda.md
index 0b4a416c6c..09cc2fddf1 100644
--- a/src/foundation/univalence.lagda.md
+++ b/src/foundation/univalence.lagda.md
@@ -148,10 +148,10 @@ compute-equiv-eq-concat :
   ((equiv-eq q) ∘e (equiv-eq p)) ＝ equiv-eq (p ∙ q)
 compute-equiv-eq-concat refl refl = eq-equiv-eq-map-equiv refl
 
-compute-eq-equiv-equiv-comp :
+compute-eq-equiv-comp-equiv :
   {l : Level} (A B C : UU l) (f : A ≃ B) (g : B ≃ C) →
   ((eq-equiv A B f) ∙ (eq-equiv B C g)) ＝ eq-equiv A C (g ∘e f)
-compute-eq-equiv-equiv-comp A B C f g =
+compute-eq-equiv-comp-equiv A B C f g =
   is-injective-map-equiv
     ( equiv-univalence)
     ( ( inv ( compute-equiv-eq-concat (eq-equiv A B f) (eq-equiv B C g))) ∙
diff --git a/src/group-theory/loop-groups-sets.lagda.md b/src/group-theory/loop-groups-sets.lagda.md
index 5aa1df7846..75f9b3f857 100644
--- a/src/group-theory/loop-groups-sets.lagda.md
+++ b/src/group-theory/loop-groups-sets.lagda.md
@@ -130,7 +130,7 @@ module _
     ( ap
       ( inv)
       ( inv
-        ( compute-eq-equiv-equiv-comp
+        ( compute-eq-equiv-comp-equiv
           ( type-Set X) (type-Set X) (type-Set X) g f))) ∙
       ( distributive-inv-concat
         ( eq-equiv (type-Set X) (type-Set X) g)

From d95a732f75b4d2299d16e70daa8ca27e5f28a591 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 11 Sep 2023 17:12:20 +0200
Subject: [PATCH 34/37] renamings

---
 ...rtier-delooping-sign-homomorphism.lagda.md | 18 ++--
 .../delooping-sign-homomorphism.lagda.md      | 10 +--
 ...mpson-delooping-sign-homomorphism.lagda.md | 18 ++--
 ...lences-families-over-subuniverses.lagda.md | 38 ++++----
 .../action-on-equivalences-functions.lagda.md | 90 +++++++++----------
 .../transport-along-equivalences.lagda.md     | 33 +++----
 6 files changed, 104 insertions(+), 103 deletions(-)

diff --git a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
index 6ee9081c2d..aa702336b4 100644
--- a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
@@ -57,7 +57,7 @@ module _
   { l : Level}
   where
 
-  not-even-difference-ap-equiv-subuniverse :
+  not-even-difference-action-equiv-family-on-subuniverse :
     (n : ℕ) (Y : 2-Element-Decidable-Subtype l (raise-Fin l (n +ℕ 2))) →
     ¬ ( sim-Equivalence-Relation
       ( even-difference-orientation-Complete-Undirected-Graph
@@ -69,7 +69,7 @@ module _
         ( star)
         ( transposition Y))
       ( map-equiv
-        ( ap-equiv-subuniverse
+        ( action-equiv-family-on-subuniverse
           ( mere-equiv-Prop (Fin (n +ℕ 2)))
           ( orientation-Complete-Undirected-Graph (n +ℕ 2))
           ( raise l (Fin (n +ℕ 2)) ,
@@ -79,7 +79,7 @@ module _
           ( transposition Y))
         ( orientation-aut-count
           (n +ℕ 2 , compute-raise l (Fin (n +ℕ 2))) star (transposition Y))))
-  not-even-difference-ap-equiv-subuniverse n =
+  not-even-difference-action-equiv-family-on-subuniverse n =
     tr
       ( λ f →
         ( Y : 2-Element-Decidable-Subtype l
@@ -110,14 +110,14 @@ module _
           preserves-id-equiv-orientation-complete-undirected-graph-equiv
             ( n +ℕ 2)}
         { y =
-          ( ap-equiv-subuniverse
+          ( action-equiv-family-on-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( orientation-Complete-Undirected-Graph (n +ℕ 2))) ,
-          ( preserves-id-equiv-ap-equiv-subuniverse
+          ( preserves-id-equiv-action-equiv-family-on-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( orientation-Complete-Undirected-Graph (n +ℕ 2)))}
         ( eq-is-contr
-          ( is-contr-preserves-id-ap-equiv-subuniverse
+          ( is-contr-preserves-id-action-equiv-family-on-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( orientation-Complete-Undirected-Graph (n +ℕ 2))
             ( is-set-orientation-Complete-Undirected-Graph (n +ℕ 2)))))
@@ -135,7 +135,7 @@ module _
       ( equiv-fin-2-quotient-sign-equiv-Fin)
       ( λ n →
         orientation-aut-count (n +ℕ 2 , compute-raise l (Fin (n +ℕ 2))) (star))
-      ( not-even-difference-ap-equiv-subuniverse)
+      ( not-even-difference-action-equiv-family-on-subuniverse)
 
   eq-cartier-delooping-sign-homomorphism :
     (n : ℕ) →
@@ -176,7 +176,7 @@ module _
               orientation-aut-count
                 ( n +ℕ 2 , compute-raise l (Fin (n +ℕ 2)))
                 ( star))
-            ( not-even-difference-ap-equiv-subuniverse)
+            ( not-even-difference-action-equiv-family-on-subuniverse)
             ( n))
           ( sign-homomorphism
             ( n +ℕ 2)
@@ -194,7 +194,7 @@ module _
       ( equiv-fin-2-quotient-sign-equiv-Fin)
       ( λ n →
         orientation-aut-count (n +ℕ 2 , compute-raise l (Fin (n +ℕ 2))) (star))
-      ( not-even-difference-ap-equiv-subuniverse)
+      ( not-even-difference-action-equiv-family-on-subuniverse)
 ```
 
 ## References
diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index 5e97d09b16..39ec19dca2 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -129,7 +129,7 @@ module _
           unit-trunc-Prop (compute-raise-Fin l1 (n +ℕ 2))))
       ( quotient-aut-succ-succ-Fin n (transposition Y))
       ( map-equiv
-        ( ap-equiv-subuniverse
+        ( action-equiv-family-on-subuniverse
           ( mere-equiv-Prop (Fin (n +ℕ 2)))
           ( D (n +ℕ 2))
           ( raise l1 (Fin (n +ℕ 2)) ,
@@ -165,13 +165,13 @@ module _
       (n : ℕ) (X X' : UU-Fin l1 n) →
       type-UU-Fin n X ≃ type-UU-Fin n X' → D n X ≃ D n X'
     invertible-action-D-equiv n =
-      ap-equiv-subuniverse (mere-equiv-Prop (Fin n)) (D n)
+      action-equiv-family-on-subuniverse (mere-equiv-Prop (Fin n)) (D n)
 
     preserves-id-equiv-invertible-action-D-equiv :
       (n : ℕ) (X : UU-Fin l1 n) →
       Id (invertible-action-D-equiv n X X id-equiv) id-equiv
     preserves-id-equiv-invertible-action-D-equiv n =
-      preserves-id-equiv-ap-equiv-subuniverse (mere-equiv-Prop (Fin n)) (D n)
+      preserves-id-equiv-action-equiv-family-on-subuniverse (mere-equiv-Prop (Fin n)) (D n)
 
     preserves-R-invertible-action-D-equiv :
       ( n : ℕ) →
@@ -184,7 +184,7 @@ module _
           ( map-equiv (invertible-action-D-equiv n X X' e) a)
           ( map-equiv (invertible-action-D-equiv n X X' e) a'))
     preserves-R-invertible-action-D-equiv n X X' =
-      Ind-ap-equiv-subuniverse (mere-equiv-Prop (Fin n)) (D n) X
+      Ind-action-equiv-family-on-subuniverse (mere-equiv-Prop (Fin n)) (D n) X
         ( λ Y f →
           ( a a' : D n X) →
           ( sim-Equivalence-Relation (R n X) a a' ↔
@@ -1488,7 +1488,7 @@ module _
             unit-trunc-Prop (compute-raise-Fin l1 (n +ℕ 2))))) →
     ¬ ( x ＝
         ( map-equiv
-          ( ap-equiv-subuniverse
+          ( action-equiv-family-on-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( type-UU-Fin 2 ∘ Q (n +ℕ 2))
             ( raise l1 (Fin (n +ℕ 2)) ,
diff --git a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
index 2f36249fd8..a27888dc68 100644
--- a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
@@ -673,7 +673,7 @@ module _
     Fin (n +ℕ 2) ≃ raise l (Fin (n +ℕ 2))
   sign-comp-aut-succ-succ-Fin n = _∘e compute-raise l (Fin (n +ℕ 2))
 
-  not-ap-equiv-subuniverse-transposition :
+  not-action-equiv-family-on-subuniverse-transposition :
     ( n : ℕ) →
     ( Y : 2-Element-Decidable-Subtype l
       ( raise-Fin l (n +ℕ 2))) →
@@ -683,7 +683,7 @@ module _
           unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2))))
       ( sign-comp-aut-succ-succ-Fin n (transposition Y))
       ( map-equiv
-        ( ap-equiv-subuniverse
+        ( action-equiv-family-on-subuniverse
           ( mere-equiv-Prop (Fin (n +ℕ 2)))
           ( λ X → Fin (n +ℕ 2) ≃ pr1 X)
           ( raise l (Fin (n +ℕ 2)) ,
@@ -692,7 +692,7 @@ module _
             unit-trunc-Prop (compute-raise-Fin l (n +ℕ 2)))
           ( transposition Y))
         ( sign-comp-aut-succ-succ-Fin n (transposition Y))))
-  not-ap-equiv-subuniverse-transposition n =
+  not-action-equiv-family-on-subuniverse-transposition n =
     tr
       ( λ f →
         ( Y : 2-Element-Decidable-Subtype l
@@ -716,14 +716,14 @@ module _
           simpson-comp-equiv (n +ℕ 2) ,
           preserves-id-equiv-simpson-comp-equiv (n +ℕ 2)}
         { y =
-          ( ap-equiv-subuniverse
+          ( action-equiv-family-on-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X) ,
-            ( preserves-id-equiv-ap-equiv-subuniverse
+            ( preserves-id-equiv-action-equiv-family-on-subuniverse
               ( mere-equiv-Prop (Fin (n +ℕ 2)))
               ( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X)))}
         ( eq-is-contr
-          ( is-contr-preserves-id-ap-equiv-subuniverse
+          ( is-contr-preserves-id-action-equiv-family-on-subuniverse
             ( mere-equiv-Prop (Fin (n +ℕ 2)))
             ( λ X → Fin (n +ℕ 2) ≃ type-UU-Fin (n +ℕ 2) X)
             ( λ X →
@@ -743,7 +743,7 @@ module _
       ( λ n _ → is-decidable-sign-comp-Equivalence-Relation n)
       ( equiv-fin-2-quotient-sign-comp-equiv-Fin)
       ( sign-comp-aut-succ-succ-Fin)
-      ( not-ap-equiv-subuniverse-transposition)
+      ( not-action-equiv-family-on-subuniverse-transposition)
 
   eq-simpson-delooping-sign-homomorphism :
     (n : ℕ) →
@@ -779,7 +779,7 @@ module _
             ( λ n H → is-decidable-sign-comp-Equivalence-Relation n)
             ( equiv-fin-2-quotient-sign-comp-equiv-Fin)
             ( sign-comp-aut-succ-succ-Fin)
-            ( not-ap-equiv-subuniverse-transposition)
+            ( not-action-equiv-family-on-subuniverse-transposition)
             ( n))
           ( sign-homomorphism
             ( n +ℕ 2)
@@ -795,7 +795,7 @@ module _
       ( λ n _ → is-decidable-sign-comp-Equivalence-Relation n)
       ( equiv-fin-2-quotient-sign-comp-equiv-Fin)
       ( sign-comp-aut-succ-succ-Fin)
-      ( not-ap-equiv-subuniverse-transposition)
+      ( not-action-equiv-family-on-subuniverse-transposition)
 ```
 
 ## References
diff --git a/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md b/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
index fae0017d26..2c5491bd07 100644
--- a/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
+++ b/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
@@ -42,18 +42,18 @@ module _
   ( P : subuniverse l1 l2) (B : type-subuniverse P → UU l3)
   where
 
-  ap-equiv-subuniverse :
+  action-equiv-family-on-subuniverse :
     (X Y : type-subuniverse P) → pr1 X ≃ pr1 Y → B X ≃ B Y
-  ap-equiv-subuniverse X Y e = equiv-tr B (eq-equiv-subuniverse P e)
+  action-equiv-family-on-subuniverse X Y e = equiv-tr B (eq-equiv-subuniverse P e)
 ```
 
 ## Properties
 
 ```agda
-  preserves-id-equiv-ap-equiv-subuniverse :
+  preserves-id-equiv-action-equiv-family-on-subuniverse :
     (X : type-subuniverse P) →
-    ap-equiv-subuniverse X X id-equiv ＝ id-equiv
-  preserves-id-equiv-ap-equiv-subuniverse X =
+    action-equiv-family-on-subuniverse X X id-equiv ＝ id-equiv
+  preserves-id-equiv-action-equiv-family-on-subuniverse X =
     ( ap (equiv-tr B)
       ( is-injective-map-equiv
         ( extensionality-subuniverse P X X)
@@ -62,39 +62,39 @@ module _
           ( id-equiv)))) ∙
       ( equiv-tr-refl B)
 
-  Ind-path-ap-equiv-subuniverse :
+  Ind-path-action-equiv-family-on-subuniverse :
     { l4 : Level}
     ( X : type-subuniverse P)
     ( F : (Y : type-subuniverse P) → B X ≃ B Y → UU l4) →
     F X id-equiv → ( Y : type-subuniverse P) (p : X ＝ Y) →
     F Y (equiv-tr B p)
-  Ind-path-ap-equiv-subuniverse X F p .X refl =
+  Ind-path-action-equiv-family-on-subuniverse X F p .X refl =
     tr (F X) (inv (equiv-tr-refl B)) p
 
-  Ind-ap-equiv-subuniverse :
+  Ind-action-equiv-family-on-subuniverse :
     { l4 : Level}
     ( X : type-subuniverse P)
     ( F : (Y : type-subuniverse P) → B X ≃ B Y → UU l4) →
     F X id-equiv → (Y : type-subuniverse P) (e : pr1 X ≃ pr1 Y) →
-    F Y (ap-equiv-subuniverse X Y e)
-  Ind-ap-equiv-subuniverse X F p Y e =
-    Ind-path-ap-equiv-subuniverse X F p Y (eq-equiv-subuniverse P e)
+    F Y (action-equiv-family-on-subuniverse X Y e)
+  Ind-action-equiv-family-on-subuniverse X F p Y e =
+    Ind-path-action-equiv-family-on-subuniverse X F p Y (eq-equiv-subuniverse P e)
 
-  is-contr-preserves-id-ap-equiv-subuniverse :
+  is-contr-preserves-id-action-equiv-family-on-subuniverse :
     ( (X : type-subuniverse P) → is-set (B X)) →
     is-contr
       ( Σ
         ( (X Y : type-subuniverse P) → pr1 X ≃ pr1 Y → B X ≃ B Y)
         ( λ D → (X : type-subuniverse P) → D X X id-equiv ＝ id-equiv))
-  pr1 (pr1 (is-contr-preserves-id-ap-equiv-subuniverse H)) =
-    ap-equiv-subuniverse
-  pr2 (pr1 (is-contr-preserves-id-ap-equiv-subuniverse H)) =
-    preserves-id-equiv-ap-equiv-subuniverse
-  pr2 (is-contr-preserves-id-ap-equiv-subuniverse H) (D , p) =
+  pr1 (pr1 (is-contr-preserves-id-action-equiv-family-on-subuniverse H)) =
+    action-equiv-family-on-subuniverse
+  pr2 (pr1 (is-contr-preserves-id-action-equiv-family-on-subuniverse H)) =
+    preserves-id-equiv-action-equiv-family-on-subuniverse
+  pr2 (is-contr-preserves-id-action-equiv-family-on-subuniverse H) (D , p) =
     eq-pair-Σ
       ( eq-htpy (λ X → eq-htpy (λ Y → eq-htpy (λ e →
-        lemma2 ap-equiv-subuniverse D
-          (λ X → preserves-id-equiv-ap-equiv-subuniverse X ∙ inv (p X))
+        lemma2 action-equiv-family-on-subuniverse D
+          (λ X → preserves-id-equiv-action-equiv-family-on-subuniverse X ∙ inv (p X))
           X Y e))))
       ( eq-is-prop
         ( is-prop-Π
diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index 91a89b5201..6bc0bc6d58 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -36,20 +36,20 @@ luckily, these two notions coincide.
 ## Definition
 
 ```agda
-eq-ap-equiv :
+action-equiv-function :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y : UU l1} →
   X ≃ Y → f X ＝ f Y
-eq-ap-equiv f {X} {Y} e = ap f (eq-equiv X Y e)
+action-equiv-function f {X} {Y} e = ap f (eq-equiv X Y e)
 
-ap-equiv :
+action-equiv-family :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
   X ≃ Y → f X ≃ f Y
-ap-equiv f = equiv-eq ∘ eq-ap-equiv f
+action-equiv-family f = equiv-eq ∘ action-equiv-function f
 
-map-ap-equiv :
+map-action-equiv-family :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
   X ≃ Y → f X → f Y
-map-ap-equiv f = map-equiv ∘ ap-equiv f
+map-action-equiv-family f = map-equiv ∘ action-equiv-family f
 ```
 
 ## Properties
@@ -57,91 +57,91 @@ map-ap-equiv f = map-equiv ∘ ap-equiv f
 ### The identity function acts trivially on equivalences
 
 ```agda
-compute-ap-equiv-id :
-  {l : Level} {X Y : UU l} (e : X ≃ Y) → (ap-equiv id e) ＝ e
-compute-ap-equiv-id {l} {X} {Y} e =
+compute-action-equiv-family-id :
+  {l : Level} {X Y : UU l} (e : X ≃ Y) → (action-equiv-family id e) ＝ e
+compute-action-equiv-family-id {l} {X} {Y} e =
   (ap equiv-eq (ap-id (eq-equiv X Y e))) ∙ (is-section-eq-equiv e)
 ```
 
 ### The action on equivalences of a composite function is the composite of the actions
 
 ```agda
-distributive-eq-ap-equiv-comp :
+distributive-action-equiv-function-comp :
   {l1 l2 l3 : Level} {C : UU l3} (g : UU l2 → C) (f : UU l1 → UU l2)
-  {X Y : UU l1} → eq-ap-equiv (g ∘ f) ~ (eq-ap-equiv g ∘ ap-equiv f)
-distributive-eq-ap-equiv-comp g f {X} {Y} e =
+  {X Y : UU l1} → action-equiv-function (g ∘ f) ~ (action-equiv-function g ∘ action-equiv-family f)
+distributive-action-equiv-function-comp g f {X} {Y} e =
   ( ap-comp g f (eq-equiv X Y e)) ∙
-  ( ap (ap g) (inv (is-retraction-eq-equiv (eq-ap-equiv f e))))
+  ( ap (ap g) (inv (is-retraction-eq-equiv (action-equiv-function f e))))
 
-distributive-ap-equiv-comp :
+distributive-action-equiv-family-comp :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2)
-  {X Y : UU l1} → ap-equiv (g ∘ f) ~ (ap-equiv g ∘ ap-equiv f)
-distributive-ap-equiv-comp g f {X} {Y} e =
-  ap equiv-eq (distributive-eq-ap-equiv-comp g f {X} {Y} e)
+  {X Y : UU l1} → action-equiv-family (g ∘ f) ~ (action-equiv-family g ∘ action-equiv-family f)
+distributive-action-equiv-family-comp g f {X} {Y} e =
+  ap equiv-eq (distributive-action-equiv-function-comp g f {X} {Y} e)
 ```
 
 ### The action on equivalences of any map preserves `id-equiv`
 
 ```agda
-compute-eq-ap-equiv-id-equiv :
+compute-action-equiv-function-id-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) (A : UU l1) →
-  (eq-ap-equiv f id-equiv) ＝ refl
-compute-eq-ap-equiv-id-equiv f A = ap (ap f) (compute-eq-equiv-id-equiv A)
+  (action-equiv-function f id-equiv) ＝ refl
+compute-action-equiv-function-id-equiv f A = ap (ap f) (compute-eq-equiv-id-equiv A)
 
-compute-ap-equiv-id-equiv :
+compute-action-equiv-family-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) (A : UU l1) →
-  (ap-equiv f id-equiv) ＝ id-equiv
-compute-ap-equiv-id-equiv f A = ap equiv-eq (compute-eq-ap-equiv-id-equiv f A)
+  (action-equiv-family f id-equiv) ＝ id-equiv
+compute-action-equiv-family-id-equiv f A = ap equiv-eq (compute-action-equiv-function-id-equiv f A)
 ```
 
 ### The action on equivalences of a constant map is constant
 
 ```agda
-compute-eq-ap-equiv-const :
+compute-action-equiv-function-const :
   {l1 l2 : Level} {B : UU l2} (b : B) {X Y : UU l1}
-  (e : X ≃ Y) → (eq-ap-equiv (const (UU l1) B b) e) ＝ refl
-compute-eq-ap-equiv-const b {X} {Y} e = ap-const b (eq-equiv X Y e)
+  (e : X ≃ Y) → (action-equiv-function (const (UU l1) B b) e) ＝ refl
+compute-action-equiv-function-const b {X} {Y} e = ap-const b (eq-equiv X Y e)
 
-compute-ap-equiv-const :
+compute-action-equiv-family-const :
   {l1 l2 : Level} (B : UU l2) {X Y : UU l1}
-  (e : X ≃ Y) → (ap-equiv (const (UU l1) (UU l2) B) e) ＝ id-equiv
-compute-ap-equiv-const B {X} {Y} e = ap equiv-eq (compute-eq-ap-equiv-const B e)
+  (e : X ≃ Y) → (action-equiv-family (const (UU l1) (UU l2) B) e) ＝ id-equiv
+compute-action-equiv-family-const B {X} {Y} e = ap equiv-eq (compute-action-equiv-function-const B e)
 ```
 
 ### The action on equivalences of any map preserves composition of equivalences
 
 ```agda
-distributive-eq-ap-equiv-comp-equiv :
+distributive-action-equiv-function-comp-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y Z : UU l1} →
   (e : X ≃ Y) (e' : Y ≃ Z) →
-  eq-ap-equiv f (e' ∘e e) ＝ (eq-ap-equiv f e ∙ eq-ap-equiv f e')
-distributive-eq-ap-equiv-comp-equiv f {X} {Y} {Z} e e' =
+  action-equiv-function f (e' ∘e e) ＝ (action-equiv-function f e ∙ action-equiv-function f e')
+distributive-action-equiv-function-comp-equiv f {X} {Y} {Z} e e' =
     ( ap (ap f) (inv (compute-eq-equiv-comp-equiv X Y Z e e'))) ∙
     ( ap-concat f (eq-equiv X Y e) (eq-equiv Y Z e'))
 
-distributive-ap-equiv-comp-equiv :
+distributive-action-equiv-family-comp-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y Z : UU l1} →
   (e : X ≃ Y) (e' : Y ≃ Z) →
-  ap-equiv f (e' ∘e e) ＝ (ap-equiv f e' ∘e ap-equiv f e)
-distributive-ap-equiv-comp-equiv f {X} {Y} {Z} e e' =
-  ( ap equiv-eq (distributive-eq-ap-equiv-comp-equiv f e e')) ∙
-  ( inv (compute-equiv-eq-concat (eq-ap-equiv f e) (eq-ap-equiv f e')))
+  action-equiv-family f (e' ∘e e) ＝ (action-equiv-family f e' ∘e action-equiv-family f e)
+distributive-action-equiv-family-comp-equiv f {X} {Y} {Z} e e' =
+  ( ap equiv-eq (distributive-action-equiv-function-comp-equiv f e e')) ∙
+  ( inv (compute-equiv-eq-concat (action-equiv-function f e) (action-equiv-function f e')))
 ```
 
 ### The action on equivalences of any map preserves inverses
 
 ```agda
-compute-eq-ap-equiv-inv-equiv :
+compute-action-equiv-function-inv-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y : UU l1}
-  (e : X ≃ Y) → eq-ap-equiv f (inv-equiv e) ＝ inv (eq-ap-equiv f e)
-compute-eq-ap-equiv-inv-equiv f {X} {Y} e =
+  (e : X ≃ Y) → action-equiv-function f (inv-equiv e) ＝ inv (action-equiv-function f e)
+compute-action-equiv-function-inv-equiv f {X} {Y} e =
   ( ap (ap f) (inv (commutativity-inv-eq-equiv X Y e))) ∙
   ( ap-inv f (eq-equiv X Y e))
 
-compute-ap-equiv-inv-equiv :
+compute-action-equiv-family-inv-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1}
-  (e : X ≃ Y) → ap-equiv f (inv-equiv e) ＝ inv-equiv (ap-equiv f e)
-compute-ap-equiv-inv-equiv f {X} {Y} e =
-  ( ap equiv-eq (compute-eq-ap-equiv-inv-equiv f e)) ∙
-  ( inv (commutativity-inv-equiv-eq (f X) (f Y) (eq-ap-equiv f e)))
+  (e : X ≃ Y) → action-equiv-family f (inv-equiv e) ＝ inv-equiv (action-equiv-family f e)
+compute-action-equiv-family-inv-equiv f {X} {Y} e =
+  ( ap equiv-eq (compute-action-equiv-function-inv-equiv f e)) ∙
+  ( inv (commutativity-inv-equiv-eq (f X) (f Y) (action-equiv-function f e)))
 ```
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 7c7d603135..96734b3409 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -146,43 +146,44 @@ eq-transpose-map-tr-equiv' f e {u} p =
 ```agda
 substitution-map-tr-equiv :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2) {X Y : UU l1}
-  (e : X ≃ Y) → map-tr-equiv g (ap-equiv f e) ~ map-tr-equiv (g ∘ f) e
+  (e : X ≃ Y) → map-tr-equiv g (action-equiv-family f e) ~ map-tr-equiv (g ∘ f) e
 substitution-map-tr-equiv g f {X} {Y} e X' =
-  ( ap (λ p → tr g p X') (is-retraction-eq-equiv (eq-ap-equiv f e))) ∙
+  ( ap (λ p → tr g p X') (is-retraction-eq-equiv (action-equiv-function f e))) ∙
   ( substitution-law-tr g f (eq-equiv X Y e))
 
 substitution-law-tr-equiv :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2) {X Y : UU l1}
-  (e : X ≃ Y) → tr-equiv g (ap-equiv f e) ＝ tr-equiv (g ∘ f) e
-substitution-law-tr-equiv g f e = eq-htpy-equiv (substitution-map-tr-equiv g f e)
+  (e : X ≃ Y) → tr-equiv g (action-equiv-family f e) ＝ tr-equiv (g ∘ f) e
+substitution-law-tr-equiv g f e =
+  eq-htpy-equiv (substitution-map-tr-equiv g f e)
 ```
 
 ### Transporting along the action on equivalences of a function
 
 ```agda
-compute-map-tr-equiv-ap-equiv :
+compute-map-tr-equiv-action-equiv-family :
   {l1 l2 l3 l4 : Level} {B : UU l1 → UU l2} {D : UU l3 → UU l4}
   (f : UU l1 → UU l3) (g : (X : UU l1) → B X → D (f X))
   {X Y : UU l1} (e : X ≃ Y) (X' : B X) →
-  map-tr-equiv D (ap-equiv f e) (g X X') ＝ g Y (map-tr-equiv B e X')
-compute-map-tr-equiv-ap-equiv {D = D} f g {X} {Y} e X' =
-  ( ap (λ p → tr D p (g X X')) (is-retraction-eq-equiv (eq-ap-equiv f e))) ∙
+  map-tr-equiv D (action-equiv-family f e) (g X X') ＝ g Y (map-tr-equiv B e X')
+compute-map-tr-equiv-action-equiv-family {D = D} f g {X} {Y} e X' =
+  ( ap (λ p → tr D p (g X X')) (is-retraction-eq-equiv (action-equiv-function f e))) ∙
   ( tr-ap f g (eq-equiv X Y e) X')
 ```
 
 ### Transport along equivalences and the action on equivalences in the universe coincide
 
 ```agda
-eq-tr-equiv-ap-equiv :
+eq-tr-equiv-action-equiv-family :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
-  (e : X ≃ Y) → tr-equiv f e ＝ ap-equiv f e
-eq-tr-equiv-ap-equiv f {X} =
+  (e : X ≃ Y) → tr-equiv f e ＝ action-equiv-family f e
+eq-tr-equiv-action-equiv-family f {X} =
   ind-equiv X
-    ( λ Y e → tr-equiv f e ＝ ap-equiv f e)
-    ( compute-tr-equiv-id-equiv f ∙ inv (compute-ap-equiv-id-equiv f X))
+    ( λ Y e → tr-equiv f e ＝ action-equiv-family f e)
+    ( compute-tr-equiv-id-equiv f ∙ inv (compute-action-equiv-family-id-equiv f X))
 
-eq-map-tr-equiv-map-ap-equiv :
+eq-map-tr-equiv-map-action-equiv-family :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
-  (e : X ≃ Y) → map-tr-equiv f e ＝ map-ap-equiv f e
-eq-map-tr-equiv-map-ap-equiv f e = ap map-equiv (eq-tr-equiv-ap-equiv f e)
+  (e : X ≃ Y) → map-tr-equiv f e ＝ map-action-equiv-family f e
+eq-map-tr-equiv-map-action-equiv-family f e = ap map-equiv (eq-tr-equiv-action-equiv-family f e)
 ```

From 8c6a1026b69766fcef1e50c783f46d1915f36847 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 11 Sep 2023 17:19:48 +0200
Subject: [PATCH 35/37] pre-commit

---
 .../delooping-sign-homomorphism.lagda.md      |  4 ++-
 ...lences-families-over-subuniverses.lagda.md |  9 +++--
 .../action-on-equivalences-functions.lagda.md | 36 +++++++++++++------
 .../transport-along-equivalences.lagda.md     | 13 ++++---
 4 files changed, 43 insertions(+), 19 deletions(-)

diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index 39ec19dca2..7830cc0611 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -171,7 +171,9 @@ module _
       (n : ℕ) (X : UU-Fin l1 n) →
       Id (invertible-action-D-equiv n X X id-equiv) id-equiv
     preserves-id-equiv-invertible-action-D-equiv n =
-      preserves-id-equiv-action-equiv-family-on-subuniverse (mere-equiv-Prop (Fin n)) (D n)
+      preserves-id-equiv-action-equiv-family-on-subuniverse
+        ( mere-equiv-Prop (Fin n))
+        ( D n)
 
     preserves-R-invertible-action-D-equiv :
       ( n : ℕ) →
diff --git a/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md b/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
index 2c5491bd07..4b5741bc23 100644
--- a/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
+++ b/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
@@ -44,7 +44,8 @@ module _
 
   action-equiv-family-on-subuniverse :
     (X Y : type-subuniverse P) → pr1 X ≃ pr1 Y → B X ≃ B Y
-  action-equiv-family-on-subuniverse X Y e = equiv-tr B (eq-equiv-subuniverse P e)
+  action-equiv-family-on-subuniverse X Y e =
+    equiv-tr B (eq-equiv-subuniverse P e)
 ```
 
 ## Properties
@@ -78,7 +79,8 @@ module _
     F X id-equiv → (Y : type-subuniverse P) (e : pr1 X ≃ pr1 Y) →
     F Y (action-equiv-family-on-subuniverse X Y e)
   Ind-action-equiv-family-on-subuniverse X F p Y e =
-    Ind-path-action-equiv-family-on-subuniverse X F p Y (eq-equiv-subuniverse P e)
+    Ind-path-action-equiv-family-on-subuniverse X F p Y
+      ( eq-equiv-subuniverse P e)
 
   is-contr-preserves-id-action-equiv-family-on-subuniverse :
     ( (X : type-subuniverse P) → is-set (B X)) →
@@ -94,7 +96,8 @@ module _
     eq-pair-Σ
       ( eq-htpy (λ X → eq-htpy (λ Y → eq-htpy (λ e →
         lemma2 action-equiv-family-on-subuniverse D
-          (λ X → preserves-id-equiv-action-equiv-family-on-subuniverse X ∙ inv (p X))
+          ( λ X →
+            preserves-id-equiv-action-equiv-family-on-subuniverse X ∙ inv (p X))
           X Y e))))
       ( eq-is-prop
         ( is-prop-Π
diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index 6bc0bc6d58..710b78bbfc 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -68,14 +68,18 @@ compute-action-equiv-family-id {l} {X} {Y} e =
 ```agda
 distributive-action-equiv-function-comp :
   {l1 l2 l3 : Level} {C : UU l3} (g : UU l2 → C) (f : UU l1 → UU l2)
-  {X Y : UU l1} → action-equiv-function (g ∘ f) ~ (action-equiv-function g ∘ action-equiv-family f)
+  {X Y : UU l1} →
+  action-equiv-function (g ∘ f) ~
+  action-equiv-function g ∘ action-equiv-family f
 distributive-action-equiv-function-comp g f {X} {Y} e =
   ( ap-comp g f (eq-equiv X Y e)) ∙
   ( ap (ap g) (inv (is-retraction-eq-equiv (action-equiv-function f e))))
 
 distributive-action-equiv-family-comp :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2)
-  {X Y : UU l1} → action-equiv-family (g ∘ f) ~ (action-equiv-family g ∘ action-equiv-family f)
+  {X Y : UU l1} →
+  action-equiv-family (g ∘ f) ~
+  action-equiv-family g ∘ action-equiv-family f
 distributive-action-equiv-family-comp g f {X} {Y} e =
   ap equiv-eq (distributive-action-equiv-function-comp g f {X} {Y} e)
 ```
@@ -86,12 +90,14 @@ distributive-action-equiv-family-comp g f {X} {Y} e =
 compute-action-equiv-function-id-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) (A : UU l1) →
   (action-equiv-function f id-equiv) ＝ refl
-compute-action-equiv-function-id-equiv f A = ap (ap f) (compute-eq-equiv-id-equiv A)
+compute-action-equiv-function-id-equiv f A =
+  ap (ap f) (compute-eq-equiv-id-equiv A)
 
 compute-action-equiv-family-id-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) (A : UU l1) →
   (action-equiv-family f id-equiv) ＝ id-equiv
-compute-action-equiv-family-id-equiv f A = ap equiv-eq (compute-action-equiv-function-id-equiv f A)
+compute-action-equiv-family-id-equiv f A =
+  ap equiv-eq (compute-action-equiv-function-id-equiv f A)
 ```
 
 ### The action on equivalences of a constant map is constant
@@ -105,7 +111,8 @@ compute-action-equiv-function-const b {X} {Y} e = ap-const b (eq-equiv X Y e)
 compute-action-equiv-family-const :
   {l1 l2 : Level} (B : UU l2) {X Y : UU l1}
   (e : X ≃ Y) → (action-equiv-family (const (UU l1) (UU l2) B) e) ＝ id-equiv
-compute-action-equiv-family-const B {X} {Y} e = ap equiv-eq (compute-action-equiv-function-const B e)
+compute-action-equiv-family-const B {X} {Y} e =
+  ap equiv-eq (compute-action-equiv-function-const B e)
 ```
 
 ### The action on equivalences of any map preserves composition of equivalences
@@ -114,7 +121,8 @@ compute-action-equiv-family-const B {X} {Y} e = ap equiv-eq (compute-action-equi
 distributive-action-equiv-function-comp-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y Z : UU l1} →
   (e : X ≃ Y) (e' : Y ≃ Z) →
-  action-equiv-function f (e' ∘e e) ＝ (action-equiv-function f e ∙ action-equiv-function f e')
+  action-equiv-function f (e' ∘e e) ＝
+  action-equiv-function f e ∙ action-equiv-function f e'
 distributive-action-equiv-function-comp-equiv f {X} {Y} {Z} e e' =
     ( ap (ap f) (inv (compute-eq-equiv-comp-equiv X Y Z e e'))) ∙
     ( ap-concat f (eq-equiv X Y e) (eq-equiv Y Z e'))
@@ -122,10 +130,14 @@ distributive-action-equiv-function-comp-equiv f {X} {Y} {Z} e e' =
 distributive-action-equiv-family-comp-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y Z : UU l1} →
   (e : X ≃ Y) (e' : Y ≃ Z) →
-  action-equiv-family f (e' ∘e e) ＝ (action-equiv-family f e' ∘e action-equiv-family f e)
-distributive-action-equiv-family-comp-equiv f {X} {Y} {Z} e e' =
+  action-equiv-family f (e' ∘e e) ＝
+  action-equiv-family f e' ∘e action-equiv-family f e
+distributive-action-equiv-family-comp-equiv f e e' =
   ( ap equiv-eq (distributive-action-equiv-function-comp-equiv f e e')) ∙
-  ( inv (compute-equiv-eq-concat (action-equiv-function f e) (action-equiv-function f e')))
+  ( inv
+    ( compute-equiv-eq-concat
+      ( action-equiv-function f e)
+      ( action-equiv-function f e')))
 ```
 
 ### The action on equivalences of any map preserves inverses
@@ -133,14 +145,16 @@ distributive-action-equiv-family-comp-equiv f {X} {Y} {Z} e e' =
 ```agda
 compute-action-equiv-function-inv-equiv :
   {l1 l2 : Level} {B : UU l2} (f : UU l1 → B) {X Y : UU l1}
-  (e : X ≃ Y) → action-equiv-function f (inv-equiv e) ＝ inv (action-equiv-function f e)
+  (e : X ≃ Y) →
+  action-equiv-function f (inv-equiv e) ＝ inv (action-equiv-function f e)
 compute-action-equiv-function-inv-equiv f {X} {Y} e =
   ( ap (ap f) (inv (commutativity-inv-eq-equiv X Y e))) ∙
   ( ap-inv f (eq-equiv X Y e))
 
 compute-action-equiv-family-inv-equiv :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1}
-  (e : X ≃ Y) → action-equiv-family f (inv-equiv e) ＝ inv-equiv (action-equiv-family f e)
+  (e : X ≃ Y) →
+  action-equiv-family f (inv-equiv e) ＝ inv-equiv (action-equiv-family f e)
 compute-action-equiv-family-inv-equiv f {X} {Y} e =
   ( ap equiv-eq (compute-action-equiv-function-inv-equiv f e)) ∙
   ( inv (commutativity-inv-equiv-eq (f X) (f Y) (action-equiv-function f e)))
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 96734b3409..58148a49ab 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -146,7 +146,8 @@ eq-transpose-map-tr-equiv' f e {u} p =
 ```agda
 substitution-map-tr-equiv :
   {l1 l2 l3 : Level} (g : UU l2 → UU l3) (f : UU l1 → UU l2) {X Y : UU l1}
-  (e : X ≃ Y) → map-tr-equiv g (action-equiv-family f e) ~ map-tr-equiv (g ∘ f) e
+  (e : X ≃ Y) →
+  map-tr-equiv g (action-equiv-family f e) ~ map-tr-equiv (g ∘ f) e
 substitution-map-tr-equiv g f {X} {Y} e X' =
   ( ap (λ p → tr g p X') (is-retraction-eq-equiv (action-equiv-function f e))) ∙
   ( substitution-law-tr g f (eq-equiv X Y e))
@@ -167,7 +168,9 @@ compute-map-tr-equiv-action-equiv-family :
   {X Y : UU l1} (e : X ≃ Y) (X' : B X) →
   map-tr-equiv D (action-equiv-family f e) (g X X') ＝ g Y (map-tr-equiv B e X')
 compute-map-tr-equiv-action-equiv-family {D = D} f g {X} {Y} e X' =
-  ( ap (λ p → tr D p (g X X')) (is-retraction-eq-equiv (action-equiv-function f e))) ∙
+  ( ap
+    ( λ p → tr D p (g X X'))
+    ( is-retraction-eq-equiv (action-equiv-function f e))) ∙
   ( tr-ap f g (eq-equiv X Y e) X')
 ```
 
@@ -180,10 +183,12 @@ eq-tr-equiv-action-equiv-family :
 eq-tr-equiv-action-equiv-family f {X} =
   ind-equiv X
     ( λ Y e → tr-equiv f e ＝ action-equiv-family f e)
-    ( compute-tr-equiv-id-equiv f ∙ inv (compute-action-equiv-family-id-equiv f X))
+    ( compute-tr-equiv-id-equiv f ∙
+      inv (compute-action-equiv-family-id-equiv f X))
 
 eq-map-tr-equiv-map-action-equiv-family :
   {l1 l2 : Level} (f : UU l1 → UU l2) {X Y : UU l1} →
   (e : X ≃ Y) → map-tr-equiv f e ＝ map-action-equiv-family f e
-eq-map-tr-equiv-map-action-equiv-family f e = ap map-equiv (eq-tr-equiv-action-equiv-family f e)
+eq-map-tr-equiv-map-action-equiv-family f e =
+  ap map-equiv (eq-tr-equiv-action-equiv-family f e)
 ```

From 1e0fab6bafd138a0be6274ed957adf20b00da1c3 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 11 Sep 2023 17:26:06 +0200
Subject: [PATCH 36/37] rename `transport` to `transport-along-identifications`
 again

---
 .../bezouts-lemma-integers.lagda.md                   |  2 +-
 .../bezouts-lemma-natural-numbers.lagda.md            |  2 +-
 src/elementary-number-theory/cofibonacci.lagda.md     |  2 +-
 .../congruence-integers.lagda.md                      |  2 +-
 .../congruence-natural-numbers.lagda.md               |  2 +-
 .../distance-natural-numbers.lagda.md                 |  2 +-
 .../divisibility-integers.lagda.md                    |  2 +-
 .../divisibility-natural-numbers.lagda.md             |  2 +-
 .../fibonacci-sequence.lagda.md                       |  2 +-
 .../fundamental-theorem-of-arithmetic.lagda.md        |  2 +-
 .../greatest-common-divisor-integers.lagda.md         |  2 +-
 .../greatest-common-divisor-natural-numbers.lagda.md  |  2 +-
 .../inequality-integers.lagda.md                      |  2 +-
 src/elementary-number-theory/powers-of-two.lagda.md   |  2 +-
 src/elementary-number-theory/prime-numbers.lagda.md   |  2 +-
 .../reduced-integer-fractions.lagda.md                |  2 +-
 .../relatively-prime-natural-numbers.lagda.md         |  2 +-
 .../strict-inequality-natural-numbers.lagda.md        |  2 +-
 ...-ordering-principle-standard-finite-types.lagda.md |  2 +-
 .../cartier-delooping-sign-homomorphism.lagda.md      |  2 +-
 .../concrete-quaternion-group.lagda.md                |  2 +-
 .../delooping-sign-homomorphism.lagda.md              |  2 +-
 src/finite-group-theory/orbits-permutations.lagda.md  |  2 +-
 .../simpson-delooping-sign-homomorphism.lagda.md      |  2 +-
 src/finite-group-theory/transpositions.lagda.md       |  2 +-
 src/foundation-core.lagda.md                          |  2 +-
 src/foundation-core/contractible-types.lagda.md       |  2 +-
 .../dependent-identifications.lagda.md                | 11 ++++++-----
 .../equality-dependent-pair-types.lagda.md            |  2 +-
 src/foundation-core/fibers-of-maps.lagda.md           |  2 +-
 .../functoriality-dependent-function-types.lagda.md   |  2 +-
 .../functoriality-dependent-pair-types.lagda.md       |  2 +-
 src/foundation-core/homotopies.lagda.md               |  2 +-
 src/foundation-core/identity-types.lagda.md           |  4 ++--
 src/foundation-core/propositions.lagda.md             |  2 +-
 src/foundation-core/singleton-induction.lagda.md      |  2 +-
 src/foundation-core/small-types.lagda.md              |  2 +-
 src/foundation-core/subtypes.lagda.md                 |  2 +-
 ...da.md => transport-along-identifications.lagda.md} |  5 +++--
 src/foundation-core/truncated-types.lagda.md          |  2 +-
 src/foundation-core/univalence.lagda.md               |  2 +-
 src/foundation.lagda.md                               |  2 +-
 ...n-equivalences-families-over-subuniverses.lagda.md |  2 +-
 .../action-on-equivalences-functions.lagda.md         |  7 ++++---
 ...on-on-identifications-dependent-functions.lagda.md |  6 +++---
 .../action-on-identifications-functions.lagda.md      |  2 +-
 ...ic-law-coproduct-and-sigma-decompositions.lagda.md |  2 +-
 ...thmetic-law-product-and-pi-decompositions.lagda.md |  2 +-
 src/foundation/binary-transport.lagda.md              |  2 +-
 src/foundation/cartesian-product-types.lagda.md       |  2 +-
 src/foundation/cones-over-cospans.lagda.md            |  2 +-
 src/foundation/coproduct-decompositions.lagda.md      |  2 +-
 src/foundation/decidable-equality.lagda.md            |  2 +-
 .../decidable-equivalence-relations.lagda.md          |  2 +-
 src/foundation/decidable-propositions.lagda.md        |  2 +-
 src/foundation/decidable-subtypes.lagda.md            |  2 +-
 src/foundation/dependent-binomial-theorem.lagda.md    |  2 +-
 src/foundation/dependent-identifications.lagda.md     |  2 +-
 src/foundation/double-negation-modality.lagda.md      |  2 +-
 .../equality-cartesian-product-types.lagda.md         |  2 +-
 src/foundation/equality-dependent-pair-types.lagda.md |  2 +-
 src/foundation/equality-fibers-of-maps.lagda.md       |  2 +-
 src/foundation/equivalence-relations.lagda.md         |  2 +-
 src/foundation/exclusive-disjunction.lagda.md         |  2 +-
 src/foundation/fiber-inclusions.lagda.md              |  2 +-
 src/foundation/full-subtypes.lagda.md                 |  2 +-
 src/foundation/function-types.lagda.md                |  2 +-
 src/foundation/functoriality-coproduct-types.lagda.md |  2 +-
 .../functoriality-dependent-function-types.lagda.md   |  2 +-
 .../functoriality-dependent-pair-types.lagda.md       |  2 +-
 src/foundation/homotopies.lagda.md                    |  2 +-
 src/foundation/identity-systems.lagda.md              |  2 +-
 src/foundation/identity-types.lagda.md                |  6 +++---
 ...uction-principle-propositional-truncation.lagda.md |  2 +-
 src/foundation/isolated-points.lagda.md               |  2 +-
 src/foundation/locally-small-types.lagda.md           |  2 +-
 src/foundation/partitions.lagda.md                    |  2 +-
 src/foundation/perfect-images.lagda.md                |  2 +-
 .../pointed-torsorial-type-families.lagda.md          |  2 +-
 .../products-unordered-pairs-of-types.lagda.md        |  2 +-
 src/foundation/propositional-extensionality.lagda.md  |  2 +-
 src/foundation/propositional-truncations.lagda.md     |  2 +-
 src/foundation/pullbacks.lagda.md                     |  2 +-
 src/foundation/singleton-induction.lagda.md           |  2 +-
 src/foundation/structure.lagda.md                     |  2 +-
 src/foundation/subuniverses.lagda.md                  |  2 +-
 src/foundation/symmetric-difference.lagda.md          |  2 +-
 src/foundation/transport-along-equivalences.lagda.md  |  9 +++++----
 ...da.md => transport-along-identifications.lagda.md} |  9 +++++----
 .../type-theoretic-principle-of-choice.lagda.md       |  2 +-
 ...nivalence-implies-function-extensionality.lagda.md |  2 +-
 src/foundation/univalent-type-families.lagda.md       |  2 +-
 src/foundation/universal-property-image.lagda.md      |  2 +-
 .../universal-property-set-quotients.lagda.md         |  2 +-
 .../equivalences-enriched-undirected-graphs.lagda.md  |  2 +-
 .../equivalences-undirected-graphs.lagda.md           |  2 +-
 src/graph-theory/fibers-directed-graphs.lagda.md      |  2 +-
 src/graph-theory/morphisms-undirected-graphs.lagda.md |  2 +-
 src/graph-theory/undirected-graphs.lagda.md           |  2 +-
 src/graph-theory/walks-directed-graphs.lagda.md       |  2 +-
 src/group-theory/concrete-group-actions.lagda.md      |  2 +-
 .../homomorphisms-concrete-group-actions.lagda.md     |  2 +-
 src/group-theory/normal-subgroups.lagda.md            |  2 +-
 src/group-theory/subgroups.lagda.md                   |  2 +-
 src/group-theory/torsors.lagda.md                     |  2 +-
 .../transitive-concrete-group-actions.lagda.md        |  2 +-
 src/higher-group-theory/higher-group-actions.lagda.md |  2 +-
 .../homomorphisms-higher-group-actions.lagda.md       |  2 +-
 src/linear-algebra/vectors.lagda.md                   |  2 +-
 src/lists/functoriality-lists.lagda.md                |  2 +-
 src/lists/permutation-lists.lagda.md                  |  2 +-
 src/lists/permutation-vectors.lagda.md                |  2 +-
 src/lists/sort-by-insertion-vectors.lagda.md          |  2 +-
 src/order-theory/finitely-graded-posets.lagda.md      |  2 +-
 src/organic-chemistry/ethane.lagda.md                 |  2 +-
 .../extensions-of-maps.lagda.md                       |  2 +-
 .../function-classes.lagda.md                         |  2 +-
 .../local-families.lagda.md                           |  2 +-
 .../local-maps.lagda.md                               |  2 +-
 .../open-modalities.lagda.md                          |  2 +-
 src/polytopes/abstract-polytopes.lagda.md             |  2 +-
 .../ideals-generated-by-subsets-rings.lagda.md        |  2 +-
 .../integer-multiples-of-elements-rings.lagda.md      |  2 +-
 .../left-ideals-generated-by-subsets-rings.lagda.md   |  2 +-
 .../right-ideals-generated-by-subsets-rings.lagda.md  |  2 +-
 src/set-theory/cumulative-hierarchy.lagda.md          |  2 +-
 ...position-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...nentials-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...products-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...nentials-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...products-species-of-types-in-subuniverses.lagda.md |  2 +-
 ...position-species-of-types-in-subuniverses.lagda.md |  2 +-
 src/species/species-of-types-in-subuniverses.lagda.md |  2 +-
 src/species/species-of-types.lagda.md                 |  2 +-
 src/structured-types/pointed-maps.lagda.md            |  2 +-
 src/synthetic-homotopy-theory/26-descent.lagda.md     |  2 +-
 src/synthetic-homotopy-theory/26-id-pushout.lagda.md  |  2 +-
 src/synthetic-homotopy-theory/circle.lagda.md         |  2 +-
 .../dependent-cocones-under-spans.lagda.md            |  2 +-
 .../dependent-pullback-property-pushouts.lagda.md     |  2 +-
 .../dependent-suspension-structures.lagda.md          |  2 +-
 .../dependent-universal-property-pushouts.lagda.md    |  2 +-
 .../dependent-universal-property-suspensions.lagda.md |  2 +-
 .../descent-circle-function-types.lagda.md            |  2 +-
 src/synthetic-homotopy-theory/descent-circle.lagda.md |  6 +++---
 .../flattening-lemma-pushouts.lagda.md                |  2 +-
 src/synthetic-homotopy-theory/free-loops.lagda.md     |  2 +-
 src/synthetic-homotopy-theory/interval-type.lagda.md  |  2 +-
 .../multiplication-circle.lagda.md                    |  2 +-
 .../sections-descent-circle.lagda.md                  |  2 +-
 .../suspension-structures.lagda.md                    |  2 +-
 .../suspensions-of-pointed-types.lagda.md             |  2 +-
 .../suspensions-of-types.lagda.md                     |  2 +-
 .../universal-cover-circle.lagda.md                   |  2 +-
 .../universal-property-circle.lagda.md                |  2 +-
 ...sal-property-suspensions-of-pointed-types.lagda.md |  2 +-
 .../universal-property-suspensions.lagda.md           |  2 +-
 src/trees/bases-directed-trees.lagda.md               |  2 +-
 src/trees/directed-trees.lagda.md                     |  2 +-
 src/trees/elementhood-relation-w-types.lagda.md       |  2 +-
 src/trees/enriched-directed-trees.lagda.md            |  2 +-
 src/trees/equivalences-directed-trees.lagda.md        |  2 +-
 .../equivalences-enriched-directed-trees.lagda.md     |  2 +-
 src/trees/extensional-w-types.lagda.md                |  2 +-
 .../functoriality-combinator-directed-trees.lagda.md  |  2 +-
 src/trees/functoriality-w-types.lagda.md              |  2 +-
 src/trees/induction-w-types.lagda.md                  |  2 +-
 src/trees/morphisms-directed-trees.lagda.md           |  2 +-
 src/trees/morphisms-enriched-directed-trees.lagda.md  |  2 +-
 src/trees/polynomial-endofunctors.lagda.md            |  2 +-
 src/trees/ranks-of-elements-w-types.lagda.md          |  2 +-
 src/trees/rooted-morphisms-directed-trees.lagda.md    |  2 +-
 .../rooted-morphisms-enriched-directed-trees.lagda.md |  2 +-
 src/trees/small-multisets.lagda.md                    |  2 +-
 ...ements-coalgebras-polynomial-endofunctors.lagda.md |  2 +-
 .../underlying-trees-of-elements-of-w-types.lagda.md  |  2 +-
 src/trees/undirected-trees.lagda.md                   |  2 +-
 src/trees/universal-multiset.lagda.md                 |  2 +-
 src/trees/w-types.lagda.md                            |  2 +-
 src/type-theories/dependent-type-theories.lagda.md    |  2 +-
 .../fibered-dependent-type-theories.lagda.md          |  2 +-
 .../sections-dependent-type-theories.lagda.md         |  2 +-
 src/type-theories/simple-type-theories.lagda.md       |  2 +-
 .../2-element-decidable-subtypes.lagda.md             |  2 +-
 .../2-element-subtypes.lagda.md                       |  2 +-
 src/univalent-combinatorics/2-element-types.lagda.md  |  2 +-
 .../counting-dependent-pair-types.lagda.md            |  2 +-
 ...cycle-prime-decomposition-natural-numbers.lagda.md |  2 +-
 .../decidable-subtypes.lagda.md                       |  2 +-
 .../dependent-pair-types.lagda.md                     |  2 +-
 .../equivalences-cubes.lagda.md                       |  2 +-
 src/univalent-combinatorics/fibers-of-maps.lagda.md   |  2 +-
 src/univalent-combinatorics/finite-types.lagda.md     |  2 +-
 .../orientations-complete-undirected-graph.lagda.md   |  2 +-
 src/univalent-combinatorics/pi-finite-types.lagda.md  |  2 +-
 src/universal-algebra/terms-over-signatures.lagda.md  |  2 +-
 196 files changed, 221 insertions(+), 216 deletions(-)
 rename src/foundation-core/{transport.lagda.md => transport-along-identifications.lagda.md} (90%)
 rename src/foundation/{transport.lagda.md => transport-along-identifications.lagda.md} (94%)

diff --git a/src/elementary-number-theory/bezouts-lemma-integers.lagda.md b/src/elementary-number-theory/bezouts-lemma-integers.lagda.md
index 536b969aa3..b94ea9b729 100644
--- a/src/elementary-number-theory/bezouts-lemma-integers.lagda.md
+++ b/src/elementary-number-theory/bezouts-lemma-integers.lagda.md
@@ -28,7 +28,7 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 ```
 
diff --git a/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md b/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md
index 19ad5b7dc7..ad127678d2 100755
--- a/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md
@@ -37,7 +37,7 @@ open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import univalent-combinatorics.standard-finite-types
diff --git a/src/elementary-number-theory/cofibonacci.lagda.md b/src/elementary-number-theory/cofibonacci.lagda.md
index e80bd5387e..8d1f91db9b 100644
--- a/src/elementary-number-theory/cofibonacci.lagda.md
+++ b/src/elementary-number-theory/cofibonacci.lagda.md
@@ -22,7 +22,7 @@ open import foundation.decidable-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/congruence-integers.lagda.md b/src/elementary-number-theory/congruence-integers.lagda.md
index 51d82b086d..9844348d01 100644
--- a/src/elementary-number-theory/congruence-integers.lagda.md
+++ b/src/elementary-number-theory/congruence-integers.lagda.md
@@ -22,7 +22,7 @@ open import foundation.binary-relations
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/congruence-natural-numbers.lagda.md b/src/elementary-number-theory/congruence-natural-numbers.lagda.md
index 732c75a31d..57781d53a3 100644
--- a/src/elementary-number-theory/congruence-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/congruence-natural-numbers.lagda.md
@@ -19,7 +19,7 @@ open import foundation.binary-relations
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import univalent-combinatorics.standard-finite-types
diff --git a/src/elementary-number-theory/distance-natural-numbers.lagda.md b/src/elementary-number-theory/distance-natural-numbers.lagda.md
index 4aa61b2b9e..0426593e35 100644
--- a/src/elementary-number-theory/distance-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/distance-natural-numbers.lagda.md
@@ -19,7 +19,7 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/divisibility-integers.lagda.md b/src/elementary-number-theory/divisibility-integers.lagda.md
index e570a1a875..b6c1fc7453 100644
--- a/src/elementary-number-theory/divisibility-integers.lagda.md
+++ b/src/elementary-number-theory/divisibility-integers.lagda.md
@@ -27,7 +27,7 @@ open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositional-maps
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md
index 97ce7e55af..869a1a4189 100644
--- a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md
@@ -23,7 +23,7 @@ open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-maps
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/elementary-number-theory/fibonacci-sequence.lagda.md b/src/elementary-number-theory/fibonacci-sequence.lagda.md
index ed4d4e50cf..634cc34706 100644
--- a/src/elementary-number-theory/fibonacci-sequence.lagda.md
+++ b/src/elementary-number-theory/fibonacci-sequence.lagda.md
@@ -17,7 +17,7 @@ open import elementary-number-theory.relatively-prime-natural-numbers
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 ```
 
 </details>
diff --git a/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md b/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md
index b05cff6d00..992aa4a2fa 100644
--- a/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md
+++ b/src/elementary-number-theory/fundamental-theorem-of-arithmetic.lagda.md
@@ -35,7 +35,7 @@ open import foundation.empty-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md b/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md
index 8fa61c143e..2dc2e9236a 100644
--- a/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md
+++ b/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md
@@ -24,7 +24,7 @@ open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md b/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md
index 32b773f17b..8b4fdaaa35 100644
--- a/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md
@@ -30,7 +30,7 @@ open import foundation.empty-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/inequality-integers.lagda.md b/src/elementary-number-theory/inequality-integers.lagda.md
index 2955f6bf44..aa92163af9 100644
--- a/src/elementary-number-theory/inequality-integers.lagda.md
+++ b/src/elementary-number-theory/inequality-integers.lagda.md
@@ -19,7 +19,7 @@ open import foundation.function-types
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/powers-of-two.lagda.md b/src/elementary-number-theory/powers-of-two.lagda.md
index d1b2c00db1..9efb3ca6ba 100644
--- a/src/elementary-number-theory/powers-of-two.lagda.md
+++ b/src/elementary-number-theory/powers-of-two.lagda.md
@@ -21,7 +21,7 @@ open import foundation.dependent-pair-types
 open import foundation.equality-cartesian-product-types
 open import foundation.function-types
 open import foundation.split-surjective-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.cartesian-product-types
diff --git a/src/elementary-number-theory/prime-numbers.lagda.md b/src/elementary-number-theory/prime-numbers.lagda.md
index ad4ce088ba..00cb8b2065 100644
--- a/src/elementary-number-theory/prime-numbers.lagda.md
+++ b/src/elementary-number-theory/prime-numbers.lagda.md
@@ -28,7 +28,7 @@ open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/reduced-integer-fractions.lagda.md b/src/elementary-number-theory/reduced-integer-fractions.lagda.md
index a6dd71808f..06aa1843e5 100644
--- a/src/elementary-number-theory/reduced-integer-fractions.lagda.md
+++ b/src/elementary-number-theory/reduced-integer-fractions.lagda.md
@@ -26,7 +26,7 @@ open import foundation.equality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md b/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md
index 61c91bb897..fbf54cb39d 100644
--- a/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md
@@ -20,7 +20,7 @@ open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md
index d238bcd5ca..6c3c272478 100644
--- a/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md
@@ -22,7 +22,7 @@ open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
diff --git a/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md b/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md
index e392a72dba..3e7b58ad9d 100644
--- a/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md
+++ b/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md
@@ -31,7 +31,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-empty-type
 open import foundation.type-arithmetic-unit-type
diff --git a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
index aa702336b4..5c0505d499 100644
--- a/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/cartier-delooping-sign-homomorphism.lagda.md
@@ -28,7 +28,7 @@ open import foundation.mere-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/finite-group-theory/concrete-quaternion-group.lagda.md b/src/finite-group-theory/concrete-quaternion-group.lagda.md
index 0c66d78790..d03507a4e9 100644
--- a/src/finite-group-theory/concrete-quaternion-group.lagda.md
+++ b/src/finite-group-theory/concrete-quaternion-group.lagda.md
@@ -13,7 +13,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.isolated-points
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import univalent-combinatorics.complements-isolated-points
diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index 7830cc0611..5e28858730 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -48,7 +48,7 @@ open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.reflecting-maps-equivalence-relations
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.uniqueness-set-quotients
 open import foundation.unit-type
diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md
index bd6ec99e84..fb5cbb2b6e 100644
--- a/src/finite-group-theory/orbits-permutations.lagda.md
+++ b/src/finite-group-theory/orbits-permutations.lagda.md
@@ -48,7 +48,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.repetitions-of-values
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
index a27888dc68..f85e3e3227 100644
--- a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md
@@ -42,7 +42,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md
index ccebbd1f04..6e2f1f1ff4 100644
--- a/src/finite-group-theory/transpositions.lagda.md
+++ b/src/finite-group-theory/transpositions.lagda.md
@@ -40,7 +40,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/foundation-core.lagda.md b/src/foundation-core.lagda.md
index db9d5bf817..6c53df4bc0 100644
--- a/src/foundation-core.lagda.md
+++ b/src/foundation-core.lagda.md
@@ -47,7 +47,7 @@ open import foundation-core.sets public
 open import foundation-core.singleton-induction public
 open import foundation-core.small-types public
 open import foundation-core.subtypes public
-open import foundation-core.transport public
+open import foundation-core.transport-along-identifications public
 open import foundation-core.truncated-maps public
 open import foundation-core.truncated-types public
 open import foundation-core.truncation-levels public
diff --git a/src/foundation-core/contractible-types.lagda.md b/src/foundation-core/contractible-types.lagda.md
index f9309add78..e56ad2bd90 100644
--- a/src/foundation-core/contractible-types.lagda.md
+++ b/src/foundation-core/contractible-types.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.equality-dependent-pair-types
 open import foundation-core.equivalences
 open import foundation-core.identity-types
 open import foundation-core.retractions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/dependent-identifications.lagda.md b/src/foundation-core/dependent-identifications.lagda.md
index 9a5da82082..55178798aa 100644
--- a/src/foundation-core/dependent-identifications.lagda.md
+++ b/src/foundation-core/dependent-identifications.lagda.md
@@ -10,7 +10,7 @@ module foundation-core.dependent-identifications where
 open import foundation.universe-levels
 
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
@@ -26,10 +26,11 @@ identification
   tr B p u ＝ v,
 ```
 
-where `tr` is the [transport](foundation-core.transport.md) function. Dependent
-identifications also satisfy groupoid laws, which are formulated appropriately
-as dependent identifications. The groupoid laws for dependent identifications
-are proven in
+where `tr` is the
+[transport](foundation-core.transport-along-identifications.md) function.
+Dependent identifications also satisfy groupoid laws, which are formulated
+appropriately as dependent identifications. The groupoid laws for dependent
+identifications are proven in
 [`foundation.dependent-identifications`](foundation.dependent-identifications.md).
 
 ## Definition
diff --git a/src/foundation-core/equality-dependent-pair-types.lagda.md b/src/foundation-core/equality-dependent-pair-types.lagda.md
index 1e4bb29d48..f80cefffdd 100644
--- a/src/foundation-core/equality-dependent-pair-types.lagda.md
+++ b/src/foundation-core/equality-dependent-pair-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/fibers-of-maps.lagda.md b/src/foundation-core/fibers-of-maps.lagda.md
index cd073f11df..788aea7aef 100644
--- a/src/foundation-core/fibers-of-maps.lagda.md
+++ b/src/foundation-core/fibers-of-maps.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/functoriality-dependent-function-types.lagda.md b/src/foundation-core/functoriality-dependent-function-types.lagda.md
index b2d10f0ab2..38efcd3954 100644
--- a/src/foundation-core/functoriality-dependent-function-types.lagda.md
+++ b/src/foundation-core/functoriality-dependent-function-types.lagda.md
@@ -26,7 +26,7 @@ open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.path-split-maps
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/functoriality-dependent-pair-types.lagda.md b/src/foundation-core/functoriality-dependent-pair-types.lagda.md
index d07321c3d6..acfba966fb 100644
--- a/src/foundation-core/functoriality-dependent-pair-types.lagda.md
+++ b/src/foundation-core/functoriality-dependent-pair-types.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.fibers-of-maps
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/homotopies.lagda.md b/src/foundation-core/homotopies.lagda.md
index 389be5b157..591cf6bcf6 100644
--- a/src/foundation-core/homotopies.lagda.md
+++ b/src/foundation-core/homotopies.lagda.md
@@ -15,7 +15,7 @@ open import foundation.universe-levels
 open import foundation-core.dependent-identifications
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/identity-types.lagda.md b/src/foundation-core/identity-types.lagda.md
index 3fb0438892..abe72f3028 100644
--- a/src/foundation-core/identity-types.lagda.md
+++ b/src/foundation-core/identity-types.lagda.md
@@ -67,8 +67,8 @@ identifications in arbitrary types.
 | Path algebra                                      | [`foundation.path-algebra`](foundation.path-algebra.md)                                                                   |
 | Symmetric identity types                          | [`foundation.symmetric-identity-types`](foundation.symmetric-identity-types.md)                                           |
 | Torsorial type families                           | [`foundation.torsorial-type-families`](foundation.torsorial-type-families.md)                                             |
-| Transport along identifications (foundation)      | [`foundation.transport`](foundation.transport.md)                                                                         |
-| Transport along identifications (foundation-core) | [`foundation-core.transport`](foundation-core.transport.md)                                                               |
+| Transport along identifications (foundation)      | [`foundation.transport-along-identifications`](foundation.transport-along-identifications.md)                             |
+| Transport along identifications (foundation-core) | [`foundation-core.transport-along-identifications`](foundation-core.transport-along-identifications.md)                   |
 | The universal property of identity types          | [`foundation.universal-property-identity-types`](foundation.universal-property-identity-types.md)                         |
 
 ## Definition
diff --git a/src/foundation-core/propositions.lagda.md b/src/foundation-core/propositions.lagda.md
index 0e54842642..4351869f15 100644
--- a/src/foundation-core/propositions.lagda.md
+++ b/src/foundation-core/propositions.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.function-extensionality
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/singleton-induction.lagda.md b/src/foundation-core/singleton-induction.lagda.md
index aa772e61a2..636f391214 100644
--- a/src/foundation-core/singleton-induction.lagda.md
+++ b/src/foundation-core/singleton-induction.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.sections
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation-core/small-types.lagda.md b/src/foundation-core/small-types.lagda.md
index ec2e2c8072..a098c9dda4 100644
--- a/src/foundation-core/small-types.lagda.md
+++ b/src/foundation-core/small-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation.logical-equivalences
 open import foundation.mere-equivalences
 open import foundation.propositional-truncations
 open import foundation.raising-universe-levels
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/foundation-core/subtypes.lagda.md b/src/foundation-core/subtypes.lagda.md
index 5ee7b28e79..316d695918 100644
--- a/src/foundation-core/subtypes.lagda.md
+++ b/src/foundation-core/subtypes.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.identity-types
 open import foundation-core.propositional-maps
 open import foundation-core.propositions
 open import foundation-core.sets
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation-core/transport.lagda.md b/src/foundation-core/transport-along-identifications.lagda.md
similarity index 90%
rename from src/foundation-core/transport.lagda.md
rename to src/foundation-core/transport-along-identifications.lagda.md
index 751cc379d2..176a78d957 100644
--- a/src/foundation-core/transport.lagda.md
+++ b/src/foundation-core/transport-along-identifications.lagda.md
@@ -1,7 +1,7 @@
 # Transport along identifications
 
 ```agda
-module foundation-core.transport where
+module foundation-core.transport-along-identifications where
 ```
 
 <details><summary>Imports</summary>
@@ -22,7 +22,8 @@ element `b : B x`, we can **transport** the element `b` along the identification
 `p` to obtain an element `tr B p b : B y`.
 
 The fact that `tr B p` is an [equivalence](foundation-core.equivalences.md) is
-recorded in [`foundation.transport`](foundation.transport.md).
+recorded in
+[`foundation.transport-along-identifications`](foundation.transport-along-identifications.md).
 
 ## Definition
 
diff --git a/src/foundation-core/truncated-types.lagda.md b/src/foundation-core/truncated-types.lagda.md
index 083ca3b4b2..bb510df533 100644
--- a/src/foundation-core/truncated-types.lagda.md
+++ b/src/foundation-core/truncated-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.retractions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 open import foundation-core.truncation-levels
 ```
 
diff --git a/src/foundation-core/univalence.lagda.md b/src/foundation-core/univalence.lagda.md
index eb72470b0f..1192ecd6e2 100644
--- a/src/foundation-core/univalence.lagda.md
+++ b/src/foundation-core/univalence.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.contractible-types
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index ddec2d8c87..3868df37de 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -256,8 +256,8 @@ open import foundation.symmetric-identity-types public
 open import foundation.symmetric-operations public
 open import foundation.tight-apartness-relations public
 open import foundation.torsorial-type-families public
-open import foundation.transport public
 open import foundation.transport-along-equivalences public
+open import foundation.transport-along-identifications public
 open import foundation.trivial-relaxed-sigma-decompositions public
 open import foundation.trivial-sigma-decompositions public
 open import foundation.truncated-equality public
diff --git a/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md b/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
index 4b5741bc23..fc3b20eb18 100644
--- a/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
+++ b/src/foundation/action-on-equivalences-families-over-subuniverses.lagda.md
@@ -12,7 +12,7 @@ open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.sets
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index 710b78bbfc..c1dd7568d9 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -29,9 +29,10 @@ to [identifications](foundation-core.identity-types.md) arising from the
 equivalences**.
 
 Alternatively, one can apply
-[transport along identifications](foundation-core.transport.md) to get
-[transport along equivalences](foundation.transport-along-equivalences.md), but
-luckily, these two notions coincide.
+[transport along identifications](foundation-core.transport-along-identifications.md)
+to get
+[transport along equivalences](foundation.transport-along-identifications-along-equivalences.md),
+but luckily, these two notions coincide.
 
 ## Definition
 
diff --git a/src/foundation/action-on-identifications-dependent-functions.lagda.md b/src/foundation/action-on-identifications-dependent-functions.lagda.md
index e639d4d709..6db5aec11d 100644
--- a/src/foundation/action-on-identifications-dependent-functions.lagda.md
+++ b/src/foundation/action-on-identifications-dependent-functions.lagda.md
@@ -21,9 +21,9 @@ Given a dependent function `f : (x : A) → B x` and an
 [identification](foundation-core.identity-types.md) `p : x ＝ y` in `A`, we
 cannot directly compare the values `f x` and `f y`, since `f x` is an element of
 type `B x` while `f y` is an element of type `B y`. However, we can
-[transport](foundation-core.transport.md) the value `f x` along `p` to obtain an
-element of type `B y` that is comparable to the value `f y`. In other words, we
-can consider the type of
+[transport](foundation-core.transport-along-identifications.md) the value `f x`
+along `p` to obtain an element of type `B y` that is comparable to the value
+`f y`. In other words, we can consider the type of
 [dependent identifications](foundation-core.dependent-identifications.md) over
 `p` from `f x` to `f y`. The **dependent action on identifications** of `f` on
 `p` is a dependent identification
diff --git a/src/foundation/action-on-identifications-functions.lagda.md b/src/foundation/action-on-identifications-functions.lagda.md
index 7ff1b0867d..4fd28c92ed 100644
--- a/src/foundation/action-on-identifications-functions.lagda.md
+++ b/src/foundation/action-on-identifications-functions.lagda.md
@@ -12,7 +12,7 @@ open import foundation.universe-levels
 open import foundation-core.constant-maps
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md b/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md
index 5c05ab8072..ee7bbc9eb8 100644
--- a/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md
+++ b/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md b/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md
index 021e304199..e15c29d921 100644
--- a/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md
+++ b/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/binary-transport.lagda.md b/src/foundation/binary-transport.lagda.md
index 5de9ca3d4f..5ac3a40225 100644
--- a/src/foundation/binary-transport.lagda.md
+++ b/src/foundation/binary-transport.lagda.md
@@ -14,7 +14,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/cartesian-product-types.lagda.md b/src/foundation/cartesian-product-types.lagda.md
index eefc3e3266..1c324495a7 100644
--- a/src/foundation/cartesian-product-types.lagda.md
+++ b/src/foundation/cartesian-product-types.lagda.md
@@ -14,7 +14,7 @@ open import foundation.subuniverses
 open import foundation.universe-levels
 
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/cones-over-cospans.lagda.md b/src/foundation/cones-over-cospans.lagda.md
index 40e0417706..dcb1e94c20 100644
--- a/src/foundation/cones-over-cospans.lagda.md
+++ b/src/foundation/cones-over-cospans.lagda.md
@@ -18,7 +18,7 @@ open import foundation-core.contractible-types
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/coproduct-decompositions.lagda.md b/src/foundation/coproduct-decompositions.lagda.md
index e836a86eb1..179923170e 100644
--- a/src/foundation/coproduct-decompositions.lagda.md
+++ b/src/foundation/coproduct-decompositions.lagda.md
@@ -32,7 +32,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.homotopies
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 
 open import univalent-combinatorics.equality-standard-finite-types
 open import univalent-combinatorics.standard-finite-types
diff --git a/src/foundation/decidable-equality.lagda.md b/src/foundation/decidable-equality.lagda.md
index 5be1de2683..c61914611a 100644
--- a/src/foundation/decidable-equality.lagda.md
+++ b/src/foundation/decidable-equality.lagda.md
@@ -28,7 +28,7 @@ open import foundation-core.injective-maps
 open import foundation-core.propositions
 open import foundation-core.retractions
 open import foundation-core.sets
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/decidable-equivalence-relations.lagda.md b/src/foundation/decidable-equivalence-relations.lagda.md
index 91c206e11e..bb50065fb8 100644
--- a/src/foundation/decidable-equivalence-relations.lagda.md
+++ b/src/foundation/decidable-equivalence-relations.lagda.md
@@ -45,7 +45,7 @@ open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/decidable-propositions.lagda.md b/src/foundation/decidable-propositions.lagda.md
index 12c92edd54..800ce600f6 100644
--- a/src/foundation/decidable-propositions.lagda.md
+++ b/src/foundation/decidable-propositions.lagda.md
@@ -34,7 +34,7 @@ open import foundation-core.propositions
 open import foundation-core.sets
 open import foundation-core.small-types
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 
 open import univalent-combinatorics.counting
 open import univalent-combinatorics.finite-types
diff --git a/src/foundation/decidable-subtypes.lagda.md b/src/foundation/decidable-subtypes.lagda.md
index ef938cc668..20472c465b 100644
--- a/src/foundation/decidable-subtypes.lagda.md
+++ b/src/foundation/decidable-subtypes.lagda.md
@@ -25,7 +25,7 @@ open import foundation-core.function-types
 open import foundation-core.identity-types
 open import foundation-core.injective-maps
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation/dependent-binomial-theorem.lagda.md b/src/foundation/dependent-binomial-theorem.lagda.md
index a502d6f2ef..e45ba02a02 100644
--- a/src/foundation/dependent-binomial-theorem.lagda.md
+++ b/src/foundation/dependent-binomial-theorem.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-dependent-function-types
 open import foundation.raising-universe-levels
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.universal-property-coproduct-types
 open import foundation.universal-property-dependent-pair-types
diff --git a/src/foundation/dependent-identifications.lagda.md b/src/foundation/dependent-identifications.lagda.md
index c94dc0698c..a2738f938a 100644
--- a/src/foundation/dependent-identifications.lagda.md
+++ b/src/foundation/dependent-identifications.lagda.md
@@ -10,7 +10,7 @@ open import foundation-core.dependent-identifications public
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.equivalences
diff --git a/src/foundation/double-negation-modality.lagda.md b/src/foundation/double-negation-modality.lagda.md
index b956347b64..c62d92e0b0 100644
--- a/src/foundation/double-negation-modality.lagda.md
+++ b/src/foundation/double-negation-modality.lagda.md
@@ -12,7 +12,7 @@ open import foundation.universe-levels
 
 open import foundation-core.function-types
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 
 open import orthogonal-factorization-systems.local-types
 open import orthogonal-factorization-systems.modal-operators
diff --git a/src/foundation/equality-cartesian-product-types.lagda.md b/src/foundation/equality-cartesian-product-types.lagda.md
index e4bba6a073..04042db816 100644
--- a/src/foundation/equality-cartesian-product-types.lagda.md
+++ b/src/foundation/equality-cartesian-product-types.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/equality-dependent-pair-types.lagda.md b/src/foundation/equality-dependent-pair-types.lagda.md
index ab452b7651..4e3a6aae09 100644
--- a/src/foundation/equality-dependent-pair-types.lagda.md
+++ b/src/foundation/equality-dependent-pair-types.lagda.md
@@ -13,7 +13,7 @@ open import foundation.action-on-identifications-dependent-functions
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-identifications
 open import foundation.dependent-pair-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.function-types
diff --git a/src/foundation/equality-fibers-of-maps.lagda.md b/src/foundation/equality-fibers-of-maps.lagda.md
index 3d2de326e7..95464bc6a0 100644
--- a/src/foundation/equality-fibers-of-maps.lagda.md
+++ b/src/foundation/equality-fibers-of-maps.lagda.md
@@ -10,7 +10,7 @@ module foundation.equality-fibers-of-maps where
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.equality-dependent-pair-types
diff --git a/src/foundation/equivalence-relations.lagda.md b/src/foundation/equivalence-relations.lagda.md
index d06d05ef3b..63a75e84f5 100644
--- a/src/foundation/equivalence-relations.lagda.md
+++ b/src/foundation/equivalence-relations.lagda.md
@@ -39,7 +39,7 @@ open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.sets
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/exclusive-disjunction.lagda.md b/src/foundation/exclusive-disjunction.lagda.md
index e52f13f134..edb218f405 100644
--- a/src/foundation/exclusive-disjunction.lagda.md
+++ b/src/foundation/exclusive-disjunction.lagda.md
@@ -33,7 +33,7 @@ open import foundation-core.functoriality-dependent-function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.identity-types
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 
 open import univalent-combinatorics.2-element-types
 open import univalent-combinatorics.equality-finite-types
diff --git a/src/foundation/fiber-inclusions.lagda.md b/src/foundation/fiber-inclusions.lagda.md
index 1c056ed907..391888d979 100644
--- a/src/foundation/fiber-inclusions.lagda.md
+++ b/src/foundation/fiber-inclusions.lagda.md
@@ -12,7 +12,7 @@ open import foundation.cones-over-cospans
 open import foundation.dependent-pair-types
 open import foundation.faithful-maps
 open import foundation.fibers-of-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/foundation/full-subtypes.lagda.md b/src/foundation/full-subtypes.lagda.md
index da27a78574..57a9714080 100644
--- a/src/foundation/full-subtypes.lagda.md
+++ b/src/foundation/full-subtypes.lagda.md
@@ -16,7 +16,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/function-types.lagda.md b/src/foundation/function-types.lagda.md
index a14e912eb0..cbafd7364a 100644
--- a/src/foundation/function-types.lagda.md
+++ b/src/foundation/function-types.lagda.md
@@ -16,7 +16,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/functoriality-coproduct-types.lagda.md b/src/foundation/functoriality-coproduct-types.lagda.md
index 9a8b67cce7..a15dcf9bfc 100644
--- a/src/foundation/functoriality-coproduct-types.lagda.md
+++ b/src/foundation/functoriality-coproduct-types.lagda.md
@@ -35,7 +35,7 @@ open import foundation-core.identity-types
 open import foundation-core.injective-maps
 open import foundation-core.negation
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/functoriality-dependent-function-types.lagda.md b/src/foundation/functoriality-dependent-function-types.lagda.md
index fd4fd354e3..992ba5daa3 100644
--- a/src/foundation/functoriality-dependent-function-types.lagda.md
+++ b/src/foundation/functoriality-dependent-function-types.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalence-extensionality
 open import foundation.equivalences
 open import foundation.function-extensionality
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
 open import foundation.universe-levels
diff --git a/src/foundation/functoriality-dependent-pair-types.lagda.md b/src/foundation/functoriality-dependent-pair-types.lagda.md
index c13a0d0134..321df44322 100644
--- a/src/foundation/functoriality-dependent-pair-types.lagda.md
+++ b/src/foundation/functoriality-dependent-pair-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.pullbacks
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/homotopies.lagda.md b/src/foundation/homotopies.lagda.md
index 79e6c497af..9603b06a46 100644
--- a/src/foundation/homotopies.lagda.md
+++ b/src/foundation/homotopies.lagda.md
@@ -24,7 +24,7 @@ open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-function-types
 open import foundation-core.functoriality-dependent-pair-types
 open import foundation-core.sections
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/identity-systems.lagda.md b/src/foundation/identity-systems.lagda.md
index da6502913b..494a0cc6f6 100644
--- a/src/foundation/identity-systems.lagda.md
+++ b/src/foundation/identity-systems.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.contractible-types
 open import foundation-core.equivalences
 open import foundation-core.identity-types
 open import foundation-core.sections
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/identity-types.lagda.md b/src/foundation/identity-types.lagda.md
index b20808d812..08c1deac5e 100644
--- a/src/foundation/identity-types.lagda.md
+++ b/src/foundation/identity-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
@@ -55,8 +55,8 @@ identifications in arbitrary types.
 | Path algebra                                      | [`foundation.path-algebra`](foundation.path-algebra.md)                                                                   |
 | Symmetric identity types                          | [`foundation.symmetric-identity-types`](foundation.symmetric-identity-types.md)                                           |
 | Torsorial type families                           | [`foundation.torsorial-type-families`](foundation.torsorial-type-families.md)                                             |
-| Transport along identifications (foundation)      | [`foundation.transport`](foundation.transport.md)                                                                         |
-| Transport along identifications (foundation-core) | [`foundation-core.transport`](foundation-core.transport.md)                                                               |
+| Transport along identifications (foundation)      | [`foundation.transport-along-identifications`](foundation.transport-along-identifications.md)                             |
+| Transport along identifications (foundation-core) | [`foundation-core.transport-along-identifications`](foundation-core.transport-along-identifications.md)                   |
 | The universal property of identity types          | [`foundation.universal-property-identity-types`](foundation.universal-property-identity-types.md)                         |
 
 ## Properties
diff --git a/src/foundation/induction-principle-propositional-truncation.lagda.md b/src/foundation/induction-principle-propositional-truncation.lagda.md
index 803ab7d6a0..dafe752efa 100644
--- a/src/foundation/induction-principle-propositional-truncation.lagda.md
+++ b/src/foundation/induction-principle-propositional-truncation.lagda.md
@@ -12,7 +12,7 @@ open import foundation.universe-levels
 
 open import foundation-core.identity-types
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/isolated-points.lagda.md b/src/foundation/isolated-points.lagda.md
index 9c4c806248..1ee694deca 100644
--- a/src/foundation/isolated-points.lagda.md
+++ b/src/foundation/isolated-points.lagda.md
@@ -35,7 +35,7 @@ open import foundation-core.injective-maps
 open import foundation-core.propositions
 open import foundation-core.sets
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/locally-small-types.lagda.md b/src/foundation/locally-small-types.lagda.md
index dfcfb2b8d9..f16321383d 100644
--- a/src/foundation/locally-small-types.lagda.md
+++ b/src/foundation/locally-small-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.small-types
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation/partitions.lagda.md b/src/foundation/partitions.lagda.md
index ad50cd08da..a59314db2b 100644
--- a/src/foundation/partitions.lagda.md
+++ b/src/foundation/partitions.lagda.md
@@ -25,7 +25,7 @@ open import foundation.small-types
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
 open import foundation.surjective-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/foundation/perfect-images.lagda.md b/src/foundation/perfect-images.lagda.md
index 35eb0ecba3..e05c373bc3 100644
--- a/src/foundation/perfect-images.lagda.md
+++ b/src/foundation/perfect-images.lagda.md
@@ -28,7 +28,7 @@ open import foundation-core.identity-types
 open import foundation-core.injective-maps
 open import foundation-core.propositional-maps
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/pointed-torsorial-type-families.lagda.md b/src/foundation/pointed-torsorial-type-families.lagda.md
index ae36b9109a..2df25c9047 100644
--- a/src/foundation/pointed-torsorial-type-families.lagda.md
+++ b/src/foundation/pointed-torsorial-type-families.lagda.md
@@ -15,7 +15,7 @@ open import foundation.locally-small-types
 open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.sorial-type-families
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.universe-levels
 
diff --git a/src/foundation/products-unordered-pairs-of-types.lagda.md b/src/foundation/products-unordered-pairs-of-types.lagda.md
index 01b4238bfb..5741e3ab94 100644
--- a/src/foundation/products-unordered-pairs-of-types.lagda.md
+++ b/src/foundation/products-unordered-pairs-of-types.lagda.md
@@ -11,7 +11,7 @@ open import foundation.dependent-pair-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-dependent-function-types
 open import foundation.symmetric-operations
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 open import foundation.unordered-pairs
 open import foundation.unordered-pairs-of-types
diff --git a/src/foundation/propositional-extensionality.lagda.md b/src/foundation/propositional-extensionality.lagda.md
index b081d7b420..fcf3c588ce 100644
--- a/src/foundation/propositional-extensionality.lagda.md
+++ b/src/foundation/propositional-extensionality.lagda.md
@@ -15,7 +15,7 @@ open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.raising-universe-levels
 open import foundation.subtype-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md
index 8342afce15..a202424801 100644
--- a/src/foundation/propositional-truncations.lagda.md
+++ b/src/foundation/propositional-truncations.lagda.md
@@ -23,7 +23,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.sets
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 open import foundation-core.truncated-types
 open import foundation-core.truncation-levels
 ```
diff --git a/src/foundation/pullbacks.lagda.md b/src/foundation/pullbacks.lagda.md
index ee18d81d39..a6de5dbca6 100644
--- a/src/foundation/pullbacks.lagda.md
+++ b/src/foundation/pullbacks.lagda.md
@@ -31,7 +31,7 @@ open import foundation-core.equality-dependent-pair-types
 open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-function-types
 open import foundation-core.propositions
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/singleton-induction.lagda.md b/src/foundation/singleton-induction.lagda.md
index 79b42d767f..ed4b0a5879 100644
--- a/src/foundation/singleton-induction.lagda.md
+++ b/src/foundation/singleton-induction.lagda.md
@@ -16,7 +16,7 @@ open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.sections
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/structure.lagda.md b/src/foundation/structure.lagda.md
index 7010ef7209..8011baf8aa 100644
--- a/src/foundation/structure.lagda.md
+++ b/src/foundation/structure.lagda.md
@@ -14,7 +14,7 @@ open import foundation.universe-levels
 open import foundation-core.equivalences
 open import foundation-core.fibers-of-maps
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/subuniverses.lagda.md b/src/foundation/subuniverses.lagda.md
index 97f5866997..5c1d434904 100644
--- a/src/foundation/subuniverses.lagda.md
+++ b/src/foundation/subuniverses.lagda.md
@@ -20,7 +20,7 @@ open import foundation-core.equivalences
 open import foundation-core.identity-types
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/symmetric-difference.lagda.md b/src/foundation/symmetric-difference.lagda.md
index 6ae37f58b1..6049da18de 100644
--- a/src/foundation/symmetric-difference.lagda.md
+++ b/src/foundation/symmetric-difference.lagda.md
@@ -21,7 +21,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.propositions
 open import foundation-core.subtypes
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 58148a49ab..566c4abf4b 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -1,7 +1,7 @@
 # Transport along equivalences
 
 ```agda
-module foundation.transport-along-equivalences where
+module foundation.transport-along-identifications-along-equivalences where
 ```
 
 <details><summary>Imports</summary>
@@ -14,7 +14,7 @@ open import foundation.equivalence-extensionality
 open import foundation.equivalence-induction
 open import foundation.equivalences
 open import foundation.function-extensionality
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
@@ -27,8 +27,9 @@ open import foundation-core.identity-types
 
 ## Idea
 
-Applying [transport along identifications](foundation-core.transport.md) to
-[identifications](foundation-core.identity-types.md) arising from the
+Applying
+[transport along identifications](foundation-core.transport-along-identifications.md)
+to [identifications](foundation-core.identity-types.md) arising from the
 [univalence axiom](foundation.univalence.md) gives us **transport along
 equivalences**.
 
diff --git a/src/foundation/transport.lagda.md b/src/foundation/transport-along-identifications.lagda.md
similarity index 94%
rename from src/foundation/transport.lagda.md
rename to src/foundation/transport-along-identifications.lagda.md
index 3d073a2711..7574aa5fe5 100644
--- a/src/foundation/transport.lagda.md
+++ b/src/foundation/transport-along-identifications.lagda.md
@@ -1,9 +1,9 @@
 # Transport along identifications
 
 ```agda
-module foundation.transport where
+module foundation.transport-along-identifications where
 
-open import foundation-core.transport public
+open import foundation-core.transport-along-identifications public
 ```
 
 <details><summary>Imports</summary>
@@ -27,8 +27,9 @@ open import foundation-core.identity-types
 
 Given a type family `B` over `A`, an
 [identification](foundation-core.identity-types.md) `p : x ＝ y` in `A` and an
-element `b : B x`, we can [**transport**](foundation-core.transport.md) the
-element `b` along the identification `p` to obtain an element `tr B p b : B y`.
+element `b : B x`, we can
+[**transport**](foundation-core.transport-along-identifications.md) the element
+`b` along the identification `p` to obtain an element `tr B p b : B y`.
 
 The fact that `tr B p` is an [equivalence](foundation-core.equivalences.md) is
 recorded in this file.
diff --git a/src/foundation/type-theoretic-principle-of-choice.lagda.md b/src/foundation/type-theoretic-principle-of-choice.lagda.md
index d463a83f92..95ed99731d 100644
--- a/src/foundation/type-theoretic-principle-of-choice.lagda.md
+++ b/src/foundation/type-theoretic-principle-of-choice.lagda.md
@@ -17,7 +17,7 @@ open import foundation-core.equivalences
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/univalence-implies-function-extensionality.lagda.md b/src/foundation/univalence-implies-function-extensionality.lagda.md
index 893439fdd8..8380f86a8c 100644
--- a/src/foundation/univalence-implies-function-extensionality.lagda.md
+++ b/src/foundation/univalence-implies-function-extensionality.lagda.md
@@ -20,7 +20,7 @@ open import foundation-core.function-extensionality
 open import foundation-core.function-types
 open import foundation-core.homotopies
 open import foundation-core.identity-types
-open import foundation-core.transport
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
diff --git a/src/foundation/univalent-type-families.lagda.md b/src/foundation/univalent-type-families.lagda.md
index 5d2e7960cc..8297c1c7a7 100644
--- a/src/foundation/univalent-type-families.lagda.md
+++ b/src/foundation/univalent-type-families.lagda.md
@@ -7,7 +7,7 @@ module foundation.univalent-type-families where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.equivalences
diff --git a/src/foundation/universal-property-image.lagda.md b/src/foundation/universal-property-image.lagda.md
index 6f57b0faf3..e1a7bdc62e 100644
--- a/src/foundation/universal-property-image.lagda.md
+++ b/src/foundation/universal-property-image.lagda.md
@@ -14,7 +14,7 @@ open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.slice
 open import foundation.surjective-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import foundation-core.contractible-maps
diff --git a/src/foundation/universal-property-set-quotients.lagda.md b/src/foundation/universal-property-set-quotients.lagda.md
index ea589810ab..c6f5f29b9a 100644
--- a/src/foundation/universal-property-set-quotients.lagda.md
+++ b/src/foundation/universal-property-set-quotients.lagda.md
@@ -25,7 +25,7 @@ open import foundation.propositional-truncations
 open import foundation.reflecting-maps-equivalence-relations
 open import foundation.sets
 open import foundation.surjective-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universal-property-image
 open import foundation.universe-levels
 
diff --git a/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md b/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md
index 309e845787..74009d5e07 100644
--- a/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md
+++ b/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md
@@ -17,7 +17,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.enriched-undirected-graphs
diff --git a/src/graph-theory/equivalences-undirected-graphs.lagda.md b/src/graph-theory/equivalences-undirected-graphs.lagda.md
index 9384fb8e0b..8b5f0a65af 100644
--- a/src/graph-theory/equivalences-undirected-graphs.lagda.md
+++ b/src/graph-theory/equivalences-undirected-graphs.lagda.md
@@ -20,7 +20,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 open import foundation.unordered-pairs
diff --git a/src/graph-theory/fibers-directed-graphs.lagda.md b/src/graph-theory/fibers-directed-graphs.lagda.md
index 2325a19107..3b7917dec1 100644
--- a/src/graph-theory/fibers-directed-graphs.lagda.md
+++ b/src/graph-theory/fibers-directed-graphs.lagda.md
@@ -14,7 +14,7 @@ open import foundation.equality-dependent-pair-types
 open import foundation.equivalences
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/graph-theory/morphisms-undirected-graphs.lagda.md b/src/graph-theory/morphisms-undirected-graphs.lagda.md
index 99a555ded8..1989176172 100644
--- a/src/graph-theory/morphisms-undirected-graphs.lagda.md
+++ b/src/graph-theory/morphisms-undirected-graphs.lagda.md
@@ -19,7 +19,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 open import foundation.unordered-pairs
 
diff --git a/src/graph-theory/undirected-graphs.lagda.md b/src/graph-theory/undirected-graphs.lagda.md
index cab82a2c23..f7b5cbd8f5 100644
--- a/src/graph-theory/undirected-graphs.lagda.md
+++ b/src/graph-theory/undirected-graphs.lagda.md
@@ -10,7 +10,7 @@ module graph-theory.undirected-graphs where
 open import foundation.cartesian-product-types
 open import foundation.dependent-pair-types
 open import foundation.equivalences
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 open import foundation.unordered-pairs
 
diff --git a/src/graph-theory/walks-directed-graphs.lagda.md b/src/graph-theory/walks-directed-graphs.lagda.md
index aaa002c9ce..ba9655d22f 100644
--- a/src/graph-theory/walks-directed-graphs.lagda.md
+++ b/src/graph-theory/walks-directed-graphs.lagda.md
@@ -24,7 +24,7 @@ open import foundation.identity-types
 open import foundation.injective-maps
 open import foundation.negation
 open import foundation.raising-universe-levels
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.directed-graphs
diff --git a/src/group-theory/concrete-group-actions.lagda.md b/src/group-theory/concrete-group-actions.lagda.md
index bc5108da26..fe407ed17e 100644
--- a/src/group-theory/concrete-group-actions.lagda.md
+++ b/src/group-theory/concrete-group-actions.lagda.md
@@ -9,7 +9,7 @@ module group-theory.concrete-group-actions where
 ```agda
 open import foundation.function-types
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.concrete-groups
diff --git a/src/group-theory/homomorphisms-concrete-group-actions.lagda.md b/src/group-theory/homomorphisms-concrete-group-actions.lagda.md
index b86eae7b09..62057b8e87 100644
--- a/src/group-theory/homomorphisms-concrete-group-actions.lagda.md
+++ b/src/group-theory/homomorphisms-concrete-group-actions.lagda.md
@@ -16,7 +16,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.concrete-group-actions
diff --git a/src/group-theory/normal-subgroups.lagda.md b/src/group-theory/normal-subgroups.lagda.md
index e15991d6ce..83ab47f2a5 100644
--- a/src/group-theory/normal-subgroups.lagda.md
+++ b/src/group-theory/normal-subgroups.lagda.md
@@ -20,7 +20,7 @@ open import foundation.large-binary-relations
 open import foundation.propositions
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.congruence-relations-groups
diff --git a/src/group-theory/subgroups.lagda.md b/src/group-theory/subgroups.lagda.md
index ce8d9ca758..9b5cac7222 100644
--- a/src/group-theory/subgroups.lagda.md
+++ b/src/group-theory/subgroups.lagda.md
@@ -25,7 +25,7 @@ open import foundation.propositions
 open import foundation.sets
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.groups
diff --git a/src/group-theory/torsors.lagda.md b/src/group-theory/torsors.lagda.md
index ef5005aa18..80d946b9c0 100644
--- a/src/group-theory/torsors.lagda.md
+++ b/src/group-theory/torsors.lagda.md
@@ -21,7 +21,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtype-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.concrete-groups
diff --git a/src/group-theory/transitive-concrete-group-actions.lagda.md b/src/group-theory/transitive-concrete-group-actions.lagda.md
index dd3eac590f..2d589c0391 100644
--- a/src/group-theory/transitive-concrete-group-actions.lagda.md
+++ b/src/group-theory/transitive-concrete-group-actions.lagda.md
@@ -21,7 +21,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.concrete-group-actions
diff --git a/src/higher-group-theory/higher-group-actions.lagda.md b/src/higher-group-theory/higher-group-actions.lagda.md
index 3e339c1a9f..92a3590986 100644
--- a/src/higher-group-theory/higher-group-actions.lagda.md
+++ b/src/higher-group-theory/higher-group-actions.lagda.md
@@ -8,7 +8,7 @@ module higher-group-theory.higher-group-actions where
 
 ```agda
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import higher-group-theory.higher-groups
diff --git a/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md b/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md
index 6d2ee91be6..5929ab0dbc 100644
--- a/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md
+++ b/src/higher-group-theory/homomorphisms-higher-group-actions.lagda.md
@@ -12,7 +12,7 @@ open import foundation.equivalences
 open import foundation.function-extensionality
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import higher-group-theory.higher-group-actions
diff --git a/src/linear-algebra/vectors.lagda.md b/src/linear-algebra/vectors.lagda.md
index 991d0a620a..ae492a1cef 100644
--- a/src/linear-algebra/vectors.lagda.md
+++ b/src/linear-algebra/vectors.lagda.md
@@ -21,7 +21,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.unit-type
diff --git a/src/lists/functoriality-lists.lagda.md b/src/lists/functoriality-lists.lagda.md
index 62b9b8f9c9..fe70813734 100644
--- a/src/lists/functoriality-lists.lagda.md
+++ b/src/lists/functoriality-lists.lagda.md
@@ -16,7 +16,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import linear-algebra.functoriality-vectors
diff --git a/src/lists/permutation-lists.lagda.md b/src/lists/permutation-lists.lagda.md
index 32fb571f8d..875d7f1700 100644
--- a/src/lists/permutation-lists.lagda.md
+++ b/src/lists/permutation-lists.lagda.md
@@ -16,7 +16,7 @@ open import foundation.dependent-pair-types
 open import foundation.equality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import linear-algebra.functoriality-vectors
diff --git a/src/lists/permutation-vectors.lagda.md b/src/lists/permutation-vectors.lagda.md
index 16572742fd..ae8d72e68a 100644
--- a/src/lists/permutation-vectors.lagda.md
+++ b/src/lists/permutation-vectors.lagda.md
@@ -22,7 +22,7 @@ open import foundation.function-types
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import linear-algebra.functoriality-vectors
diff --git a/src/lists/sort-by-insertion-vectors.lagda.md b/src/lists/sort-by-insertion-vectors.lagda.md
index 2f9c6c74e3..6f756f9386 100644
--- a/src/lists/sort-by-insertion-vectors.lagda.md
+++ b/src/lists/sort-by-insertion-vectors.lagda.md
@@ -18,7 +18,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/order-theory/finitely-graded-posets.lagda.md b/src/order-theory/finitely-graded-posets.lagda.md
index 2b684e743a..7534d942df 100644
--- a/src/order-theory/finitely-graded-posets.lagda.md
+++ b/src/order-theory/finitely-graded-posets.lagda.md
@@ -24,7 +24,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/organic-chemistry/ethane.lagda.md b/src/organic-chemistry/ethane.lagda.md
index 74afce7451..a284da7eff 100644
--- a/src/organic-chemistry/ethane.lagda.md
+++ b/src/organic-chemistry/ethane.lagda.md
@@ -23,7 +23,7 @@ open import foundation.identity-types
 open import foundation.injective-maps
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 open import foundation.unordered-pairs
diff --git a/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md b/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md
index aabd238396..adab6dcfbc 100644
--- a/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md
+++ b/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md
@@ -26,7 +26,7 @@ open import foundation.monomorphisms
 open import foundation.propositions
 open import foundation.sets
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-arithmetic-dependent-pair-types
diff --git a/src/orthogonal-factorization-systems/function-classes.lagda.md b/src/orthogonal-factorization-systems/function-classes.lagda.md
index 56e789dbf9..9c30c83a67 100644
--- a/src/orthogonal-factorization-systems/function-classes.lagda.md
+++ b/src/orthogonal-factorization-systems/function-classes.lagda.md
@@ -14,7 +14,7 @@ open import foundation.function-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.pullback-squares
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 ```
diff --git a/src/orthogonal-factorization-systems/local-families.lagda.md b/src/orthogonal-factorization-systems/local-families.lagda.md
index eeea9e9704..fa92bac793 100644
--- a/src/orthogonal-factorization-systems/local-families.lagda.md
+++ b/src/orthogonal-factorization-systems/local-families.lagda.md
@@ -23,7 +23,7 @@ open import foundation.identity-types
 open import foundation.propositions
 open import foundation.retractions
 open import foundation.sections
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-function-types
 open import foundation.type-arithmetic-unit-type
 open import foundation.unit-type
diff --git a/src/orthogonal-factorization-systems/local-maps.lagda.md b/src/orthogonal-factorization-systems/local-maps.lagda.md
index cf854ead40..14a6b71b46 100644
--- a/src/orthogonal-factorization-systems/local-maps.lagda.md
+++ b/src/orthogonal-factorization-systems/local-maps.lagda.md
@@ -23,7 +23,7 @@ open import foundation.identity-types
 open import foundation.propositions
 open import foundation.retractions
 open import foundation.sections
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-function-types
 open import foundation.type-arithmetic-unit-type
 open import foundation.unit-type
diff --git a/src/orthogonal-factorization-systems/open-modalities.lagda.md b/src/orthogonal-factorization-systems/open-modalities.lagda.md
index 5e1ffc17e0..45a170fa4c 100644
--- a/src/orthogonal-factorization-systems/open-modalities.lagda.md
+++ b/src/orthogonal-factorization-systems/open-modalities.lagda.md
@@ -14,7 +14,7 @@ open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.locally-small-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import orthogonal-factorization-systems.higher-modalities
diff --git a/src/polytopes/abstract-polytopes.lagda.md b/src/polytopes/abstract-polytopes.lagda.md
index 02b69a6446..a490cdeffd 100644
--- a/src/polytopes/abstract-polytopes.lagda.md
+++ b/src/polytopes/abstract-polytopes.lagda.md
@@ -17,7 +17,7 @@ open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md
index e550494571..9350b7af52 100644
--- a/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md
+++ b/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md
@@ -17,7 +17,7 @@ open import foundation.logical-equivalences
 open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unions-subtypes
 open import foundation.universe-levels
 
diff --git a/src/ring-theory/integer-multiples-of-elements-rings.lagda.md b/src/ring-theory/integer-multiples-of-elements-rings.lagda.md
index 6dae1f60d1..5f2d652de3 100644
--- a/src/ring-theory/integer-multiples-of-elements-rings.lagda.md
+++ b/src/ring-theory/integer-multiples-of-elements-rings.lagda.md
@@ -15,7 +15,7 @@ open import elementary-number-theory.natural-numbers
 open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md
index 8ae0db4466..2eebe34272 100644
--- a/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md
+++ b/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md
@@ -18,7 +18,7 @@ open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unions-subtypes
 open import foundation.universe-levels
 
diff --git a/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md
index c0819f3d2f..38d5ceeffb 100644
--- a/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md
+++ b/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md
@@ -18,7 +18,7 @@ open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unions-subtypes
 open import foundation.universe-levels
 
diff --git a/src/set-theory/cumulative-hierarchy.lagda.md b/src/set-theory/cumulative-hierarchy.lagda.md
index a499a76770..6ce0932a89 100644
--- a/src/set-theory/cumulative-hierarchy.lagda.md
+++ b/src/set-theory/cumulative-hierarchy.lagda.md
@@ -27,7 +27,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md
index 8ddb767df3..3a83b52b38 100644
--- a/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md
@@ -22,7 +22,7 @@ open import foundation.relaxed-sigma-decompositions
 open import foundation.sigma-closed-subuniverses
 open import foundation.sigma-decomposition-subuniverse
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-theoretic-principle-of-choice
diff --git a/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md
index fd7f92f200..f308e753bc 100644
--- a/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md
@@ -22,7 +22,7 @@ open import foundation.relaxed-sigma-decompositions
 open import foundation.sigma-closed-subuniverses
 open import foundation.sigma-decomposition-subuniverse
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.unit-type
 open import foundation.univalence
diff --git a/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md
index 9c146364e4..03ad6b59e3 100644
--- a/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md
@@ -20,7 +20,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
diff --git a/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md b/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md
index 6cc067707a..d581585b1f 100644
--- a/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md
@@ -20,7 +20,7 @@ open import foundation.pi-decompositions-subuniverse
 open import foundation.product-decompositions
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.univalence
 open import foundation.universe-levels
diff --git a/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md b/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md
index 6a966673f9..71bdd90718 100644
--- a/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md
@@ -19,7 +19,7 @@ open import foundation.product-decompositions
 open import foundation.product-decompositions-subuniverse
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
diff --git a/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md b/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md
index 11c167bc6e..f848a96a4f 100644
--- a/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md
+++ b/src/species/small-cauchy-composition-species-of-types-in-subuniverses.lagda.md
@@ -22,7 +22,7 @@ open import foundation.sigma-closed-subuniverses
 open import foundation.sigma-decomposition-subuniverse
 open import foundation.small-types
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-theoretic-principle-of-choice
diff --git a/src/species/species-of-types-in-subuniverses.lagda.md b/src/species/species-of-types-in-subuniverses.lagda.md
index 890975d154..1ef1c85fec 100644
--- a/src/species/species-of-types-in-subuniverses.lagda.md
+++ b/src/species/species-of-types-in-subuniverses.lagda.md
@@ -13,7 +13,7 @@ open import foundation.equivalences
 open import foundation.function-types
 open import foundation.propositions
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/species/species-of-types.lagda.md b/src/species/species-of-types.lagda.md
index 585fc15a8c..ceb687541f 100644
--- a/src/species/species-of-types.lagda.md
+++ b/src/species/species-of-types.lagda.md
@@ -9,7 +9,7 @@ module species.species-of-types where
 ```agda
 open import foundation.cartesian-product-types
 open import foundation.equivalences
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 ```
diff --git a/src/structured-types/pointed-maps.lagda.md b/src/structured-types/pointed-maps.lagda.md
index 76986cc813..ec9b8563f6 100644
--- a/src/structured-types/pointed-maps.lagda.md
+++ b/src/structured-types/pointed-maps.lagda.md
@@ -13,7 +13,7 @@ open import foundation.constant-maps
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import structured-types.pointed-dependent-functions
diff --git a/src/synthetic-homotopy-theory/26-descent.lagda.md b/src/synthetic-homotopy-theory/26-descent.lagda.md
index 0db3a07d53..b5f0f3c13d 100644
--- a/src/synthetic-homotopy-theory/26-descent.lagda.md
+++ b/src/synthetic-homotopy-theory/26-descent.lagda.md
@@ -30,7 +30,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.pullbacks
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-theoretic-principle-of-choice
 open import foundation.univalence
 open import foundation.universe-levels
diff --git a/src/synthetic-homotopy-theory/26-id-pushout.lagda.md b/src/synthetic-homotopy-theory/26-id-pushout.lagda.md
index dd8365d954..1b7e25ccc4 100644
--- a/src/synthetic-homotopy-theory/26-id-pushout.lagda.md
+++ b/src/synthetic-homotopy-theory/26-id-pushout.lagda.md
@@ -22,7 +22,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universal-property-identity-types
 open import foundation.universe-levels
 
diff --git a/src/synthetic-homotopy-theory/circle.lagda.md b/src/synthetic-homotopy-theory/circle.lagda.md
index b33a747e72..71067c794a 100644
--- a/src/synthetic-homotopy-theory/circle.lagda.md
+++ b/src/synthetic-homotopy-theory/circle.lagda.md
@@ -16,7 +16,7 @@ open import foundation.identity-types
 open import foundation.mere-equality
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import structured-types.pointed-types
diff --git a/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md b/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md
index 1c6f3f5ea4..fd04ca5e1f 100644
--- a/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md
@@ -19,7 +19,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.cocones-under-spans
diff --git a/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md b/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md
index 96775f12df..2017650ebe 100644
--- a/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-pullback-property-pushouts.lagda.md
@@ -12,7 +12,7 @@ open import foundation.cones-over-cospans
 open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.pullbacks
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.cocones-under-spans
diff --git a/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md b/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md
index 5eebd4008d..a2096ced2b 100644
--- a/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-suspension-structures.lagda.md
@@ -24,7 +24,7 @@ open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.injective-maps
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md b/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md
index c59d13aae4..073a8f2feb 100644
--- a/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md
@@ -20,7 +20,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.pullbacks
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.cocones-under-spans
diff --git a/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md b/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md
index d3bca48c18..fd85921d9e 100644
--- a/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md
+++ b/src/synthetic-homotopy-theory/dependent-universal-property-suspensions.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md b/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md
index 15f9bbe907..4a56bcd36b 100644
--- a/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md
+++ b/src/synthetic-homotopy-theory/descent-circle-function-types.lagda.md
@@ -17,7 +17,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.functoriality-function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.descent-circle
diff --git a/src/synthetic-homotopy-theory/descent-circle.lagda.md b/src/synthetic-homotopy-theory/descent-circle.lagda.md
index 0c85533f60..a99bdc3c45 100644
--- a/src/synthetic-homotopy-theory/descent-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/descent-circle.lagda.md
@@ -25,7 +25,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.univalence
 open import foundation.universe-levels
@@ -121,8 +121,8 @@ module _
 ### Canonical descent data for a family over the circle
 
 A type family over the circle gives rise to its canonical descent data, obtained
-by evaluation at `base` and [transporting](foundation-core.transport.md) along
-`loop`.
+by evaluation at `base` and
+[transporting](foundation-core.transport-along-identifications.md) along `loop`.
 
 ```agda
 ev-descent-data-circle :
diff --git a/src/synthetic-homotopy-theory/flattening-lemma-pushouts.lagda.md b/src/synthetic-homotopy-theory/flattening-lemma-pushouts.lagda.md
index e8a9010429..0280f9eac0 100644
--- a/src/synthetic-homotopy-theory/flattening-lemma-pushouts.lagda.md
+++ b/src/synthetic-homotopy-theory/flattening-lemma-pushouts.lagda.md
@@ -11,7 +11,7 @@ open import foundation.commuting-squares-of-maps
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.functoriality-dependent-pair-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.cocones-under-spans
diff --git a/src/synthetic-homotopy-theory/free-loops.lagda.md b/src/synthetic-homotopy-theory/free-loops.lagda.md
index 5e83bb9173..9fe6f480e3 100644
--- a/src/synthetic-homotopy-theory/free-loops.lagda.md
+++ b/src/synthetic-homotopy-theory/free-loops.lagda.md
@@ -16,7 +16,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/synthetic-homotopy-theory/interval-type.lagda.md b/src/synthetic-homotopy-theory/interval-type.lagda.md
index 52c072070b..1492581343 100644
--- a/src/synthetic-homotopy-theory/interval-type.lagda.md
+++ b/src/synthetic-homotopy-theory/interval-type.lagda.md
@@ -15,7 +15,7 @@ open import foundation.equivalences
 open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/synthetic-homotopy-theory/multiplication-circle.lagda.md b/src/synthetic-homotopy-theory/multiplication-circle.lagda.md
index 4fb8d2102a..9e1ac9c52c 100644
--- a/src/synthetic-homotopy-theory/multiplication-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/multiplication-circle.lagda.md
@@ -13,7 +13,7 @@ open import foundation.function-extensionality
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import structured-types.pointed-homotopies
diff --git a/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md b/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md
index e1d3cac8c3..6db8ae32e0 100644
--- a/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/sections-descent-circle.lagda.md
@@ -23,7 +23,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.path-algebra
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.descent-circle
diff --git a/src/synthetic-homotopy-theory/suspension-structures.lagda.md b/src/synthetic-homotopy-theory/suspension-structures.lagda.md
index 58cf49bf18..51084e96be 100644
--- a/src/synthetic-homotopy-theory/suspension-structures.lagda.md
+++ b/src/synthetic-homotopy-theory/suspension-structures.lagda.md
@@ -25,7 +25,7 @@ open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.injective-maps
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md b/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md
index af23cf6315..66d7735858 100644
--- a/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md
+++ b/src/synthetic-homotopy-theory/suspensions-of-pointed-types.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md b/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md
index 8779ed0686..517d08df40 100644
--- a/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md
+++ b/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md
@@ -24,7 +24,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md
index 5b4d1b9f23..940f31faf7 100644
--- a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md
@@ -29,7 +29,7 @@ open import foundation.identity-types
 open import foundation.raising-universe-levels
 open import foundation.sets
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.universe-levels
diff --git a/src/synthetic-homotopy-theory/universal-property-circle.lagda.md b/src/synthetic-homotopy-theory/universal-property-circle.lagda.md
index 71abc58f61..769e039acb 100644
--- a/src/synthetic-homotopy-theory/universal-property-circle.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-property-circle.lagda.md
@@ -22,7 +22,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sections
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import synthetic-homotopy-theory.free-loops
diff --git a/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md b/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
index 5379230ecb..aa892fc08a 100644
--- a/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-property-suspensions-of-pointed-types.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md b/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md
index 77cd4006aa..6974a7afa2 100644
--- a/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md
+++ b/src/synthetic-homotopy-theory/universal-property-suspensions.lagda.md
@@ -23,7 +23,7 @@ open import foundation.homotopies
 open import foundation.identity-systems
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universal-property-unit-type
diff --git a/src/trees/bases-directed-trees.lagda.md b/src/trees/bases-directed-trees.lagda.md
index fa49199117..62493b1ade 100644
--- a/src/trees/bases-directed-trees.lagda.md
+++ b/src/trees/bases-directed-trees.lagda.md
@@ -21,7 +21,7 @@ open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
 open import foundation.universe-levels
diff --git a/src/trees/directed-trees.lagda.md b/src/trees/directed-trees.lagda.md
index 8c72d592d4..629fb4c5bb 100644
--- a/src/trees/directed-trees.lagda.md
+++ b/src/trees/directed-trees.lagda.md
@@ -25,7 +25,7 @@ open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
diff --git a/src/trees/elementhood-relation-w-types.lagda.md b/src/trees/elementhood-relation-w-types.lagda.md
index 4205e542be..df5c81a9b4 100644
--- a/src/trees/elementhood-relation-w-types.lagda.md
+++ b/src/trees/elementhood-relation-w-types.lagda.md
@@ -10,7 +10,7 @@ module trees.elementhood-relation-w-types where
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.elementhood-relation-coalgebras-polynomial-endofunctors
diff --git a/src/trees/enriched-directed-trees.lagda.md b/src/trees/enriched-directed-trees.lagda.md
index 57fbf3744d..570286771d 100644
--- a/src/trees/enriched-directed-trees.lagda.md
+++ b/src/trees/enriched-directed-trees.lagda.md
@@ -20,7 +20,7 @@ open import foundation.identity-types
 open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.directed-graphs
diff --git a/src/trees/equivalences-directed-trees.lagda.md b/src/trees/equivalences-directed-trees.lagda.md
index 89badad851..c508da1874 100644
--- a/src/trees/equivalences-directed-trees.lagda.md
+++ b/src/trees/equivalences-directed-trees.lagda.md
@@ -19,7 +19,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.subtype-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.equivalences-directed-graphs
diff --git a/src/trees/equivalences-enriched-directed-trees.lagda.md b/src/trees/equivalences-enriched-directed-trees.lagda.md
index e24e0e65d7..c796443d76 100644
--- a/src/trees/equivalences-enriched-directed-trees.lagda.md
+++ b/src/trees/equivalences-enriched-directed-trees.lagda.md
@@ -20,7 +20,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.enriched-directed-trees
diff --git a/src/trees/extensional-w-types.lagda.md b/src/trees/extensional-w-types.lagda.md
index d53f7b4c1a..3a8887dc5d 100644
--- a/src/trees/extensional-w-types.lagda.md
+++ b/src/trees/extensional-w-types.lagda.md
@@ -22,7 +22,7 @@ open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.slice
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.univalent-type-families
 open import foundation.universe-levels
diff --git a/src/trees/functoriality-combinator-directed-trees.lagda.md b/src/trees/functoriality-combinator-directed-trees.lagda.md
index b181907762..62ecdf2b9c 100644
--- a/src/trees/functoriality-combinator-directed-trees.lagda.md
+++ b/src/trees/functoriality-combinator-directed-trees.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.combinator-directed-trees
diff --git a/src/trees/functoriality-w-types.lagda.md b/src/trees/functoriality-w-types.lagda.md
index d2327b6b22..6a32e1a93e 100644
--- a/src/trees/functoriality-w-types.lagda.md
+++ b/src/trees/functoriality-w-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.functoriality-dependent-function-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositional-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-maps
 open import foundation.truncated-types
 open import foundation.truncation-levels
diff --git a/src/trees/induction-w-types.lagda.md b/src/trees/induction-w-types.lagda.md
index bf924b6e13..1805b55311 100644
--- a/src/trees/induction-w-types.lagda.md
+++ b/src/trees/induction-w-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation.function-extensionality
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.elementhood-relation-w-types
diff --git a/src/trees/morphisms-directed-trees.lagda.md b/src/trees/morphisms-directed-trees.lagda.md
index f4cb1d987a..eea276a027 100644
--- a/src/trees/morphisms-directed-trees.lagda.md
+++ b/src/trees/morphisms-directed-trees.lagda.md
@@ -17,7 +17,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.negation
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-empty-type
 open import foundation.universe-levels
 
diff --git a/src/trees/morphisms-enriched-directed-trees.lagda.md b/src/trees/morphisms-enriched-directed-trees.lagda.md
index ea3167e9b1..c6618ae770 100644
--- a/src/trees/morphisms-enriched-directed-trees.lagda.md
+++ b/src/trees/morphisms-enriched-directed-trees.lagda.md
@@ -14,7 +14,7 @@ open import foundation.dependent-pair-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.enriched-directed-trees
diff --git a/src/trees/polynomial-endofunctors.lagda.md b/src/trees/polynomial-endofunctors.lagda.md
index 96be64542a..d2d8d3280d 100644
--- a/src/trees/polynomial-endofunctors.lagda.md
+++ b/src/trees/polynomial-endofunctors.lagda.md
@@ -16,7 +16,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/trees/ranks-of-elements-w-types.lagda.md b/src/trees/ranks-of-elements-w-types.lagda.md
index 4ef53cb626..1efe2269d1 100644
--- a/src/trees/ranks-of-elements-w-types.lagda.md
+++ b/src/trees/ranks-of-elements-w-types.lagda.md
@@ -15,7 +15,7 @@ open import foundation.identity-types
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.elementhood-relation-w-types
diff --git a/src/trees/rooted-morphisms-directed-trees.lagda.md b/src/trees/rooted-morphisms-directed-trees.lagda.md
index 2b46217d61..aad2dd69aa 100644
--- a/src/trees/rooted-morphisms-directed-trees.lagda.md
+++ b/src/trees/rooted-morphisms-directed-trees.lagda.md
@@ -18,7 +18,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subtype-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.bases-directed-trees
diff --git a/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md b/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md
index 1836a44f6d..f876ff518a 100644
--- a/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md
+++ b/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md
@@ -12,7 +12,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.enriched-directed-trees
diff --git a/src/trees/small-multisets.lagda.md b/src/trees/small-multisets.lagda.md
index 3388a2abad..33175f9ad2 100644
--- a/src/trees/small-multisets.lagda.md
+++ b/src/trees/small-multisets.lagda.md
@@ -18,7 +18,7 @@ open import foundation.identity-types
 open import foundation.propositions
 open import foundation.small-types
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md b/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md
index 9b9453ec89..0c7f181af9 100644
--- a/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md
+++ b/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md
@@ -23,7 +23,7 @@ open import foundation.identity-types
 open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-empty-type
 open import foundation.universe-levels
 
diff --git a/src/trees/underlying-trees-of-elements-of-w-types.lagda.md b/src/trees/underlying-trees-of-elements-of-w-types.lagda.md
index 2a196afe79..6579f766cd 100644
--- a/src/trees/underlying-trees-of-elements-of-w-types.lagda.md
+++ b/src/trees/underlying-trees-of-elements-of-w-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.identity-types
 open import foundation.isolated-points
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.directed-graphs
diff --git a/src/trees/undirected-trees.lagda.md b/src/trees/undirected-trees.lagda.md
index 539065d90d..b9317cfec2 100644
--- a/src/trees/undirected-trees.lagda.md
+++ b/src/trees/undirected-trees.lagda.md
@@ -20,7 +20,7 @@ open import foundation.mere-equality
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import graph-theory.paths-undirected-graphs
diff --git a/src/trees/universal-multiset.lagda.md b/src/trees/universal-multiset.lagda.md
index 126ca1365c..274f1dfe83 100644
--- a/src/trees/universal-multiset.lagda.md
+++ b/src/trees/universal-multiset.lagda.md
@@ -13,7 +13,7 @@ open import foundation.identity-types
 open import foundation.raising-universe-levels
 open import foundation.small-types
 open import foundation.small-universes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import trees.functoriality-w-types
diff --git a/src/trees/w-types.lagda.md b/src/trees/w-types.lagda.md
index 76777c1305..eeae83e192 100644
--- a/src/trees/w-types.lagda.md
+++ b/src/trees/w-types.lagda.md
@@ -19,7 +19,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-theoretic-principle-of-choice
diff --git a/src/type-theories/dependent-type-theories.lagda.md b/src/type-theories/dependent-type-theories.lagda.md
index 9fb662749e..3c9f43e8b1 100644
--- a/src/type-theories/dependent-type-theories.lagda.md
+++ b/src/type-theories/dependent-type-theories.lagda.md
@@ -17,7 +17,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/type-theories/fibered-dependent-type-theories.lagda.md b/src/type-theories/fibered-dependent-type-theories.lagda.md
index fef7eb8f07..a5da36d53a 100644
--- a/src/type-theories/fibered-dependent-type-theories.lagda.md
+++ b/src/type-theories/fibered-dependent-type-theories.lagda.md
@@ -13,7 +13,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import type-theories.dependent-type-theories
diff --git a/src/type-theories/sections-dependent-type-theories.lagda.md b/src/type-theories/sections-dependent-type-theories.lagda.md
index 57a5c18782..339dc4c6e7 100644
--- a/src/type-theories/sections-dependent-type-theories.lagda.md
+++ b/src/type-theories/sections-dependent-type-theories.lagda.md
@@ -10,7 +10,7 @@ module type-theories.sections-dependent-type-theories where
 
 ```agda
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import type-theories.dependent-type-theories
diff --git a/src/type-theories/simple-type-theories.lagda.md b/src/type-theories/simple-type-theories.lagda.md
index 227dc224ff..98461a1a5e 100644
--- a/src/type-theories/simple-type-theories.lagda.md
+++ b/src/type-theories/simple-type-theories.lagda.md
@@ -13,7 +13,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
diff --git a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md
index 4f925227c3..335deed2c7 100644
--- a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md
+++ b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md
@@ -34,7 +34,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.univalence
 open import foundation.universe-levels
diff --git a/src/univalent-combinatorics/2-element-subtypes.lagda.md b/src/univalent-combinatorics/2-element-subtypes.lagda.md
index 5ba14b8c22..d41430d2f4 100644
--- a/src/univalent-combinatorics/2-element-subtypes.lagda.md
+++ b/src/univalent-combinatorics/2-element-subtypes.lagda.md
@@ -24,7 +24,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.unit-type
 open import foundation.universe-levels
diff --git a/src/univalent-combinatorics/2-element-types.lagda.md b/src/univalent-combinatorics/2-element-types.lagda.md
index ae036a1d18..68e6cf1568 100644
--- a/src/univalent-combinatorics/2-element-types.lagda.md
+++ b/src/univalent-combinatorics/2-element-types.lagda.md
@@ -37,7 +37,7 @@ open import foundation.propositions
 open import foundation.raising-universe-levels
 open import foundation.sets
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
diff --git a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md
index 7da51e828b..a6e6581dd9 100644
--- a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md
+++ b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md
@@ -26,7 +26,7 @@ open import foundation.functoriality-coproduct-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.sections
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-unit-type
diff --git a/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md b/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md
index f916a53857..a93706de19 100644
--- a/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md
+++ b/src/univalent-combinatorics/cycle-prime-decomposition-natural-numbers.lagda.md
@@ -22,7 +22,7 @@ open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.iterated-cartesian-product-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universe-levels
 
diff --git a/src/univalent-combinatorics/decidable-subtypes.lagda.md b/src/univalent-combinatorics/decidable-subtypes.lagda.md
index 90a64f18aa..66b79ec69f 100644
--- a/src/univalent-combinatorics/decidable-subtypes.lagda.md
+++ b/src/univalent-combinatorics/decidable-subtypes.lagda.md
@@ -22,7 +22,7 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import univalent-combinatorics.decidable-dependent-pair-types
diff --git a/src/univalent-combinatorics/dependent-pair-types.lagda.md b/src/univalent-combinatorics/dependent-pair-types.lagda.md
index cf8877984e..f0e9351cf6 100644
--- a/src/univalent-combinatorics/dependent-pair-types.lagda.md
+++ b/src/univalent-combinatorics/dependent-pair-types.lagda.md
@@ -25,7 +25,7 @@ open import foundation.propositions
 open import foundation.sections
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-coproduct-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
diff --git a/src/univalent-combinatorics/equivalences-cubes.lagda.md b/src/univalent-combinatorics/equivalences-cubes.lagda.md
index 89f968dd7a..f71497fe95 100644
--- a/src/univalent-combinatorics/equivalences-cubes.lagda.md
+++ b/src/univalent-combinatorics/equivalences-cubes.lagda.md
@@ -19,7 +19,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.structure-identity-principle
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import univalent-combinatorics.cubes
diff --git a/src/univalent-combinatorics/fibers-of-maps.lagda.md b/src/univalent-combinatorics/fibers-of-maps.lagda.md
index 1e80f22eaa..1779e20173 100644
--- a/src/univalent-combinatorics/fibers-of-maps.lagda.md
+++ b/src/univalent-combinatorics/fibers-of-maps.lagda.md
@@ -21,7 +21,7 @@ open import foundation.identity-types
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sections
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 
diff --git a/src/univalent-combinatorics/finite-types.lagda.md b/src/univalent-combinatorics/finite-types.lagda.md
index 2791564474..4fa2bac999 100644
--- a/src/univalent-combinatorics/finite-types.lagda.md
+++ b/src/univalent-combinatorics/finite-types.lagda.md
@@ -32,7 +32,7 @@ open import foundation.raising-universe-levels
 open import foundation.sets
 open import foundation.subtypes
 open import foundation.subuniverses
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-arithmetic-empty-type
 open import foundation.unit-type
diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md
index c964610d74..97718030aa 100644
--- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md
+++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md
@@ -48,7 +48,7 @@ open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 
diff --git a/src/univalent-combinatorics/pi-finite-types.lagda.md b/src/univalent-combinatorics/pi-finite-types.lagda.md
index d2bbfc548b..1ee3a422e5 100644
--- a/src/univalent-combinatorics/pi-finite-types.lagda.md
+++ b/src/univalent-combinatorics/pi-finite-types.lagda.md
@@ -46,7 +46,7 @@ open import foundation.set-truncations
 open import foundation.sets
 open import foundation.subtypes
 open import foundation.surjective-maps
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-arithmetic-coproduct-types
diff --git a/src/universal-algebra/terms-over-signatures.lagda.md b/src/universal-algebra/terms-over-signatures.lagda.md
index 48f19c048d..8a5cf6febf 100644
--- a/src/universal-algebra/terms-over-signatures.lagda.md
+++ b/src/universal-algebra/terms-over-signatures.lagda.md
@@ -14,7 +14,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.identity-types
-open import foundation.transport
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 

From 8ee37e1d04c5ebb5d6f45298e8c3efbe3197ed0f Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 11 Sep 2023 17:33:45 +0200
Subject: [PATCH 37/37] fix a search and replace mistake

---
 src/foundation/action-on-equivalences-functions.lagda.md | 4 ++--
 src/foundation/transport-along-equivalences.lagda.md     | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/foundation/action-on-equivalences-functions.lagda.md b/src/foundation/action-on-equivalences-functions.lagda.md
index c1dd7568d9..1fa32b1f30 100644
--- a/src/foundation/action-on-equivalences-functions.lagda.md
+++ b/src/foundation/action-on-equivalences-functions.lagda.md
@@ -31,8 +31,8 @@ equivalences**.
 Alternatively, one can apply
 [transport along identifications](foundation-core.transport-along-identifications.md)
 to get
-[transport along equivalences](foundation.transport-along-identifications-along-equivalences.md),
-but luckily, these two notions coincide.
+[transport along equivalences](foundation.transport-along-equivalences.md), but
+luckily, these two notions coincide.
 
 ## Definition
 
diff --git a/src/foundation/transport-along-equivalences.lagda.md b/src/foundation/transport-along-equivalences.lagda.md
index 566c4abf4b..ebf2304b5c 100644
--- a/src/foundation/transport-along-equivalences.lagda.md
+++ b/src/foundation/transport-along-equivalences.lagda.md
@@ -1,7 +1,7 @@
 # Transport along equivalences
 
 ```agda
-module foundation.transport-along-identifications-along-equivalences where
+module foundation.transport-along-equivalences where
 ```
 
 <details><summary>Imports</summary>