diff --git a/src/sage/categories/tensor.py b/src/sage/categories/tensor.py
index 01208e6217e..c36eb11a5ff 100644
--- a/src/sage/categories/tensor.py
+++ b/src/sage/categories/tensor.py
@@ -13,10 +13,11 @@
 # ****************************************************************************
 
 from sage.categories.covariant_functorial_construction import CovariantFunctorialConstruction, CovariantConstructionCategory
+from sage.categories.pushout import MultivariateConstructionFunctor
 from sage.typeset.unicode_characters import unicode_otimes
 
 
-class TensorProductFunctor(CovariantFunctorialConstruction):
+class TensorProductFunctor(CovariantFunctorialConstruction, MultivariateConstructionFunctor):
     """
     A singleton class for the tensor functor.
 
@@ -51,7 +52,24 @@ class TensorProductFunctor(CovariantFunctorialConstruction):
     _functor_category = "TensorProducts"
     symbol = " # "
     unicode_symbol = f" {unicode_otimes} "
+    def __init__(self, category=None):
+        r"""
+        Constructor. See :class:`TensorProductFunctor` for details.
 
+        TESTS::
+
+            sage: from sage.categories.tensor import TensorProductFunctor
+            sage: TensorProductFunctor()
+            The tensor functorial construction
+        """
+        CovariantFunctorialConstruction.__init__(self)
+        self._forced_category = category
+        from sage.categories.rings import Rings
+        if self._forced_category is not None:
+            codomain = self._forced_category
+        else:
+            codomain = Rings()
+        MultivariateConstructionFunctor.__init__(self, Rings(), codomain)
 
 tensor = TensorProductFunctor()
 
diff --git a/src/sage/combinat/free_module.py b/src/sage/combinat/free_module.py
index 8a97fbdc796..0988afb29cf 100644
--- a/src/sage/combinat/free_module.py
+++ b/src/sage/combinat/free_module.py
@@ -503,10 +503,13 @@ def change_ring(self, R):
             return self
         construction = self.construction()
         if construction is not None:
-            functor, _ = construction
+            functor, args = construction
             from sage.categories.pushout import VectorFunctor
+            from sage.categories.tensor import TensorProductFunctor
             if isinstance(functor, VectorFunctor):
                 return functor(R)
+            if isinstance(functor, TensorProductFunctor):
+                return functor([f.change_ring(R) for f in args])
         raise NotImplementedError('the method change_ring() has not yet been implemented')
 
     # For backwards compatibility
@@ -1259,8 +1262,8 @@ class CombinatorialFreeModule_Tensor(CombinatorialFreeModule):
             Category of tensor products of
              finite dimensional modules with basis over Integer Ring
 
-            sage: T.construction() # todo: not implemented
-            [tensor, ]
+            sage: T.construction()
+            (The tensor functorial construction, (F, G))
 
         T is a free module, with same base ring as F and G::
 
@@ -1347,7 +1350,8 @@ def __classcall_private__(cls, modules, **options):
             assert (all(module in ModulesWithBasis(R)) for module in modules)
             # should check the base ring
             # flatten the list of modules so that tensor(A, tensor(B,C)) gets rewritten into tensor(A, B, C)
-            modules = sum([module._sets if isinstance(module, CombinatorialFreeModule_Tensor) else (module,) for module in modules], ())
+            modules = sum([module._sets if isinstance(module, CombinatorialFreeModule_Tensor) else (module,)
+                           for module in modules], ())
             if all('FiniteDimensional' in M.category().axioms() for M in modules):
                 options['category'] = options['category'].FiniteDimensional()
             return super(CombinatorialFreeModule.Tensor, cls).__classcall__(cls, modules, **options)