diff --git a/src/lightning/fabric/accelerators/registry.py b/src/lightning/fabric/accelerators/registry.py
index 4959a0fb9426a..539b7aa8a01dc 100644
--- a/src/lightning/fabric/accelerators/registry.py
+++ b/src/lightning/fabric/accelerators/registry.py
@@ -73,14 +73,14 @@ def register(
         data["description"] = description
         data["init_params"] = init_params
 
-        def do_register(name: str, accelerator: Callable) -> Callable:
+        def do_register(accelerator: Callable) -> Callable:
             data["accelerator"] = accelerator
             data["accelerator_name"] = name
             self[name] = data
             return accelerator
 
         if accelerator is not None:
-            return do_register(name, accelerator)
+            return do_register(accelerator)
 
         return do_register