Handle singledispatch functions with rewritten signatures.

If a singledispatch function has its `__signature__` rewritten, autodoc
fails to annotate that signature because `func.__annotations__` is not
consulted.  Instead, directly assign to `__signature__` instead.

sphinx/ext/autodoc/__init__.py

@@ -13,6 +13,7 @@
 import importlib
 import re
 import warnings
+from inspect import Parameter
 from types import ModuleType
 from typing import Any, Callable, Dict, Iterator, List, Sequence, Set, Tuple, Type, Union
 from unittest.mock import patch
@@ -1108,9 +1109,10 @@ class SingledispatchFunctionDocumenter(FunctionDocumenter):
         if len(sig.parameters) == 0:
             return
 
-        name = list(sig.parameters)[0]
-        if name not in func.__annotations__:
-            func.__annotations__[name] = typ
+        params = list(sig.parameters.values())
+        if params[0].annotation is Parameter.empty:
+            params[0] = params[0].replace(annotation=typ)
+            func.__signature__ = sig.replace(parameters=params)  # type: ignore
 
 
 class DecoratorDocumenter(FunctionDocumenter):
@@ -1508,13 +1510,14 @@ class SingledispatchMethodDocumenter(MethodDocumenter):
 
     def annotate_to_first_argument(self, func: Callable, typ: Type) -> None:
         """Annotate type hint to the first argument of function if needed."""
-        sig = inspect.signature(func, bound_method=True)
-        if len(sig.parameters) == 0:
+        sig = inspect.signature(func)
+        if len(sig.parameters) == 1:
             return
 
-        name = list(sig.parameters)[0]
-        if name not in func.__annotations__:
-            func.__annotations__[name] = typ
+        params = list(sig.parameters.values())
+        if params[1].annotation is Parameter.empty:
+            params[1] = params[1].replace(annotation=typ)
+            func.__signature__ = sig.replace(parameters=params)  # type: ignore
 
 
 class AttributeDocumenter(DocstringStripSignatureMixin, ClassLevelDocumenter):  # type: ignore

tests/roots/test-ext-autodoc/target/singledispatch.py

@@ -1,4 +1,11 @@
 from functools import singledispatch
+import inspect
+
+
+def assign_signature(func):
+    # This is intended to cover more complex signature-rewriting decorators.
+    func.__signature__ = inspect.signature(func)
+    return func
 
 
 @singledispatch
@@ -14,6 +21,7 @@ def _func_int(arg, kwarg=None):
 
 
 @func.register(str)
+@assign_signature
 def _func_str(arg, kwarg=None):
     """A function for str."""
     pass