diff --git a/stdlib/3/functools.pyi b/stdlib/3/functools.pyi
index 562c32bcd..10c46f42e 100644
--- a/stdlib/3/functools.pyi
+++ b/stdlib/3/functools.pyi
@@ -88,10 +88,17 @@ class partialmethod(Generic[_T]):
 class _SingleDispatchCallable(Generic[_T]):
     registry: Mapping[Any, Callable[..., _T]]
     def dispatch(self, cls: Any) -> Callable[..., _T]: ...
+    # @fun.register(complex)
+    # def _(arg, verbose=False): ...
     @overload
-    def register(self, cls: Any) -> Callable[[Callable[..., _T]], Callable[..., _T]]: ...
+    def register(self, cls: type, func: None = ...) -> Callable[[Callable[..., _T]], Callable[..., _T]]: ...
+    # @fun.register
+    # def _(arg: int, verbose=False):
     @overload
-    def register(self, cls: Any, func: Callable[..., _T]) -> Callable[..., _T]: ...
+    def register(self, cls: Callable[..., _T], func: None = ...) -> Callable[..., _T]: ...
+    # fun.register(int, lambda x: x)
+    @overload
+    def register(self, cls: Type, func: Callable[..., _T]) -> Callable[..., _T]: ...
     def _clear_cache(self) -> None: ...
     def __call__(self, *args: Any, **kwargs: Any) -> _T: ...
 
@@ -103,9 +110,11 @@ if sys.version_info >= (3, 8):
         func: Callable[..., _T]
         def __init__(self, func: Callable[..., _T]) -> None: ...
         @overload
-        def register(self, cls: Any, method: None = ...) -> Callable[[Callable[..., _T]], Callable[..., _T]]: ...
+        def register(self, cls: Type, method: None = ...) -> Callable[[Callable[..., _T]], Callable[..., _T]]: ...
         @overload
-        def register(self, cls: Any, method: Callable[..., _T]) -> Callable[..., _T]: ...
+        def register(self, cls: Callable[..., _T], method: None = ...) -> Callable[..., _T]: ...
+        @overload
+        def register(self, cls: Type, method: Callable[..., _T]) -> Callable[..., _T]: ...
         def __call__(self, *args: Any, **kwargs: Any) -> _T: ...
     class cached_property(Generic[_T]):
         func: Callable[[Any], _T]