diff --git a/pytato/transform/__init__.py b/pytato/transform/__init__.py
index 942cf4f48..ca93a8187 100644
--- a/pytato/transform/__init__.py
+++ b/pytato/transform/__init__.py
@@ -38,6 +38,7 @@
     TypeAlias,
     TypeVar,
     cast,
+    overload,
 )
 
 import numpy as np
@@ -262,10 +263,36 @@ def rec_function_definition(
         assert method is not None
         return method(expr, *args, **kwargs)
 
-    def __call__(self,
-                 expr: ArrayOrNames, *args: P.args, **kwargs: P.kwargs) -> ResultT:
+    @overload
+    def __call__(
+            self,
+            expr: ArrayOrNames,
+            *args: P.args,
+            **kwargs: P.kwargs) -> ResultT:
+        ...
+
+    @overload
+    def __call__(
+            self,
+            expr: FunctionDefinition,
+            *args: P.args,
+            **kwargs: P.kwargs) -> FunctionResultT:
+        ...
+
+    def __call__(
+            self,
+            expr: ArrayOrNames | FunctionDefinition,
+            *args: P.args,
+            **kwargs: P.kwargs) -> ResultT | FunctionResultT:
         """Handle the mapping of *expr*."""
-        return self.rec(expr, *args, **kwargs)
+        if isinstance(expr, ArrayOrNames):
+            return self.rec(expr, *args, **kwargs)
+        elif isinstance(expr, FunctionDefinition):
+            return self.rec_function_definition(expr, *args, **kwargs)
+        else:
+            raise ForeignObjectError(
+                f"{type(self).__name__} encountered invalid foreign "
+                f"object: {expr!r}") from None
 
 # }}}
 
@@ -1847,7 +1874,7 @@ def __init__(self) -> None:
         self.node_to_users: dict[ArrayOrNames,
                 set[DistributedSend | ArrayOrNames]] = {}
 
-    def __call__(self, expr: ArrayOrNames) -> None:
+    def __call__(self, expr: ArrayOrNames) -> None:  # type: ignore[override]
         # Root node has no predecessor
         self.node_to_users[expr] = set()
         self.rec(expr)
diff --git a/pytato/transform/lower_to_index_lambda.py b/pytato/transform/lower_to_index_lambda.py
index 205d578e7..0b91c716f 100644
--- a/pytato/transform/lower_to_index_lambda.py
+++ b/pytato/transform/lower_to_index_lambda.py
@@ -688,7 +688,8 @@ def handle_unsupported_array(self, expr: Array) -> Array:
     def rec(self, expr: Array) -> Array:  # type: ignore[override]
         return expr
 
-    __call__ = Mapper.rec
+    def __call__(self, expr: Array) -> Array:  # type: ignore[override]
+        return Mapper.rec(self, expr)
 
 
 def to_index_lambda(expr: Array) -> IndexLambda: