diff --git a/libsolidity/analysis/DeclarationTypeChecker.cpp b/libsolidity/analysis/DeclarationTypeChecker.cpp
index e2f0cf6e8b00..4515203d9603 100644
--- a/libsolidity/analysis/DeclarationTypeChecker.cpp
+++ b/libsolidity/analysis/DeclarationTypeChecker.cpp
@@ -143,17 +143,35 @@ void DeclarationTypeChecker::endVisit(UserDefinedValueTypeDefinition const& _use
 {
 	TypeName const* typeName = _userDefined.underlyingType();
 	solAssert(typeName, "");
-	if (!dynamic_cast<ElementaryTypeName const*>(typeName))
+
+	if (
+		!dynamic_cast<ElementaryTypeName const*>(typeName) &&
+		!dynamic_cast<FunctionTypeName const*>(typeName)
+	)
 		m_errorReporter.fatalTypeError(
 			8657_error,
 			typeName->location(),
-			"The underlying type for a user defined value type has to be an elementary value type."
+			"The underlying type for a user defined value type has to be an elementary value type or an internal function type."
 		);
 
 	Type const* type = typeName->annotation().type;
 	solAssert(type, "");
 	solAssert(!dynamic_cast<UserDefinedValueType const*>(type), "");
-	if (!type->isValueType())
+
+	if (auto const* funType = dynamic_cast<FunctionType const*>(type))
+	{
+		if (funType->kind() != FunctionType::Kind::Internal)
+			m_errorReporter.fatalTypeError(
+				8657_error,
+				typeName->location(),
+				"The underlying type of the user defined value type \"" +
+				_userDefined.name() +
+				"\" has to be an internal function type, but \"" +
+				type->toString(true) +
+				"\" is an external function type."
+			);
+	}
+	else if (!type->isValueType())
 		m_errorReporter.typeError(
 			8129_error,
 			_userDefined.location(),
diff --git a/test/libsolidity/semanticTests/userDefinedValueType/function_type_internal.sol b/test/libsolidity/semanticTests/userDefinedValueType/function_type_internal.sol
new file mode 100644
index 000000000000..d264761d4b0b
--- /dev/null
+++ b/test/libsolidity/semanticTests/userDefinedValueType/function_type_internal.sol
@@ -0,0 +1,38 @@
+library Hooks {
+    type Callback is function(uint256) returns (uint256);
+}
+
+contract C {
+    Hooks.Callback cb;
+
+    function double(uint256 x) internal pure returns (uint256) {
+        return x * 2;
+    }
+
+    function triple(uint256 x) internal pure returns (uint256) {
+        return x * 3;
+    }
+
+    function setCb(Hooks.Callback _cb) internal {
+        cb = _cb;
+    }
+
+    function setup() public {
+        setCb(Hooks.Callback.wrap(double));
+    }
+
+    function callCb(uint256 x) public returns (uint256) {
+        function(uint256) returns (uint256) fn = Hooks.Callback.unwrap(cb);
+        return fn(x);
+    }
+
+    function wrapAndCall(uint256 x) public returns (uint256) {
+        Hooks.Callback wrapped = Hooks.Callback.wrap(triple);
+        function(uint256) returns (uint256) fn = Hooks.Callback.unwrap(wrapped);
+        return fn(x);
+    }
+}
+// ----
+// setup() ->
+// callCb(uint256): 5 -> 10
+// wrapAndCall(uint256): 7 -> 21
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/all_value_types.sol b/test/libsolidity/syntaxTests/userDefinedValueType/all_value_types.sol
index e1d453007ae8..be327671a8cf 100644
--- a/test/libsolidity/syntaxTests/userDefinedValueType/all_value_types.sol
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/all_value_types.sol
@@ -9,5 +9,7 @@ type MyUfixed is ufixed;
 type MyFixedBytes32 is bytes32;
 type MyFixedBytes1 is bytes1;
 type MyBool is bool;
+type MyCallback is function(uint256) returns (bool);
+type MyPureFunction is function(uint256, uint256) pure returns (uint256);
 /// test to see if having NatSpec causes issues
 type redundantNatSpec is bytes2;
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/forward_reference_err.sol b/test/libsolidity/syntaxTests/userDefinedValueType/forward_reference_err.sol
index 9ea93cb0902c..3c1dc091ed13 100644
--- a/test/libsolidity/syntaxTests/userDefinedValueType/forward_reference_err.sol
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/forward_reference_err.sol
@@ -1,4 +1,4 @@
 function f(MyIntB x) pure {}
 type MyIntB is MyIntB;
 // ----
-// TypeError 8657: (44-50): The underlying type for a user defined value type has to be an elementary value type.
+// TypeError 8657: (44-50): The underlying type for a user defined value type has to be an elementary value type or an internal function type.
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/function_type_internal.sol b/test/libsolidity/syntaxTests/userDefinedValueType/function_type_internal.sol
new file mode 100644
index 000000000000..7093bfd7828c
--- /dev/null
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/function_type_internal.sol
@@ -0,0 +1,27 @@
+type Callback is function(uint256) returns (bool);
+
+library Hooks {
+    type BeforeSwap is function(uint256) returns (bool);
+}
+
+contract C {
+    Hooks.BeforeSwap cb;
+
+    function set(Hooks.BeforeSwap _cb) internal {
+        cb = _cb;
+    }
+
+    function wrapUnwrap() internal pure {
+        function(uint256) returns (bool) raw;
+        Callback wrapped = Callback.wrap(raw);
+        function(uint256) returns (bool) unwrapped = Callback.unwrap(wrapped);
+        unwrapped;
+    }
+
+    function wrapUnwrapLib() internal pure {
+        function(uint256) returns (bool) raw;
+        Hooks.BeforeSwap wrapped = Hooks.BeforeSwap.wrap(raw);
+        function(uint256) returns (bool) unwrapped = Hooks.BeforeSwap.unwrap(wrapped);
+        unwrapped;
+    }
+}
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_contract_err.sol b/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_contract_err.sol
index 5a86e92f996a..fbebd810a9eb 100644
--- a/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_contract_err.sol
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_contract_err.sol
@@ -1,4 +1,4 @@
 contract C {}
 type MyContract is C;
 // ----
-// TypeError 8657: (33-34): The underlying type for a user defined value type has to be an elementary value type.
+// TypeError 8657: (33-34): The underlying type for a user defined value type has to be an elementary value type or an internal function type.
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_enum_err.sol b/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_enum_err.sol
index f143bc509312..3a757f5a7a7a 100644
--- a/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_enum_err.sol
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_enum_err.sol
@@ -2,4 +2,4 @@ enum E {A, B, C}
 
 type MyType is E;
 // ----
-// TypeError 8657: (33-34): The underlying type for a user defined value type has to be an elementary value type.
+// TypeError 8657: (33-34): The underlying type for a user defined value type has to be an elementary value type or an internal function type.
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_function_err.sol b/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_function_err.sol
index 78d9c70431e2..37253318314a 100644
--- a/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_function_err.sol
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_function_err.sol
@@ -1,3 +1,3 @@
-type MyFunction is function(uint) returns (uint);
+type MyFunction is function(uint) external returns (uint);
 // ----
-// TypeError 8657: (19-49): The underlying type for a user defined value type has to be an elementary value type.
+// TypeError 8657: (19-58): The underlying type of the user defined value type "MyFunction" has to be an internal function type, but "function (uint256) external returns (uint256)" is an external function type.
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_mapping_err.sol b/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_mapping_err.sol
index 9f894c521b80..068ea5de113c 100644
--- a/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_mapping_err.sol
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_mapping_err.sol
@@ -1,3 +1,3 @@
 type MyInt is mapping(uint => uint);
 // ----
-// TypeError 8657: (14-35): The underlying type for a user defined value type has to be an elementary value type.
+// TypeError 8657: (14-35): The underlying type for a user defined value type has to be an elementary value type or an internal function type.
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_struct_err.sol b/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_struct_err.sol
index 3a713a887691..14c7c44c1542 100644
--- a/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_struct_err.sol
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/non_value_type_struct_err.sol
@@ -4,4 +4,4 @@ contract C {
     type MyType is S;
 }
 // ----
-// TypeError 8657: (52-53): The underlying type for a user defined value type has to be an elementary value type.
+// TypeError 8657: (52-53): The underlying type for a user defined value type has to be an elementary value type or an internal function type.
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/recursive_err.sol b/test/libsolidity/syntaxTests/userDefinedValueType/recursive_err.sol
index 4130edde5d9a..8fd3dd1e067f 100644
--- a/test/libsolidity/syntaxTests/userDefinedValueType/recursive_err.sol
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/recursive_err.sol
@@ -1,4 +1,4 @@
 type MyInt1 is MyInt2;
 type MyInt2 is MyInt1;
 // ----
-// TypeError 8657: (15-21): The underlying type for a user defined value type has to be an elementary value type.
+// TypeError 8657: (15-21): The underlying type for a user defined value type has to be an elementary value type or an internal function type.
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/recursive_function_parameter_err.sol b/test/libsolidity/syntaxTests/userDefinedValueType/recursive_function_parameter_err.sol
index 52e99311e4ad..771efd3062ff 100644
--- a/test/libsolidity/syntaxTests/userDefinedValueType/recursive_function_parameter_err.sol
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/recursive_function_parameter_err.sol
@@ -1,3 +1,3 @@
 type MyFunction is function(MyFunction) external returns(MyFunction);
 // ----
-// TypeError 8657: (19-69): The underlying type for a user defined value type has to be an elementary value type.
+// TypeError 8657: (19-69): The underlying type of the user defined value type "MyFunction" has to be an internal function type, but "function (MyFunction) external returns (MyFunction)" is an external function type.
diff --git a/test/libsolidity/syntaxTests/userDefinedValueType/self_reference_err.sol b/test/libsolidity/syntaxTests/userDefinedValueType/self_reference_err.sol
index 20c657c7341a..08ef1cca3285 100644
--- a/test/libsolidity/syntaxTests/userDefinedValueType/self_reference_err.sol
+++ b/test/libsolidity/syntaxTests/userDefinedValueType/self_reference_err.sol
@@ -1,3 +1,3 @@
 type MyInt is MyInt;
 // ----
-// TypeError 8657: (14-19): The underlying type for a user defined value type has to be an elementary value type.
+// TypeError 8657: (14-19): The underlying type for a user defined value type has to be an elementary value type or an internal function type.