diff --git a/src/sage/manifolds/differentiable/manifold.py b/src/sage/manifolds/differentiable/manifold.py
index 7fa589be987..d650d4eb712 100644
--- a/src/sage/manifolds/differentiable/manifold.py
+++ b/src/sage/manifolds/differentiable/manifold.py
@@ -1438,9 +1438,9 @@ def tensor_field_module(self, tensor_type, dest_map=None):
             Free module T^(2,1)(U) of type-(2,1) tensors fields on the Open
              subset U of the 3-dimensional differentiable manifold M
             sage: TU.category()
-            Category of finite dimensional modules over Algebra of
-             differentiable scalar fields on the Open subset U of the
-             3-dimensional differentiable manifold M
+            Category of tensor products of finite dimensional modules
+             over Algebra of differentiable scalar fields
+              on the Open subset U of the 3-dimensional differentiable manifold M
             sage: TU.base_ring()
             Algebra of differentiable scalar fields on the Open subset U of
              the 3-dimensional differentiable manifold M
diff --git a/src/sage/manifolds/differentiable/tensorfield_module.py b/src/sage/manifolds/differentiable/tensorfield_module.py
index d68264a472c..cbc73f9e520 100644
--- a/src/sage/manifolds/differentiable/tensorfield_module.py
+++ b/src/sage/manifolds/differentiable/tensorfield_module.py
@@ -652,8 +652,8 @@ class TensorFieldFreeModule(TensorFreeModule):
     `T^{(2,0)}(\RR^3)` is a module over the algebra `C^k(\RR^3)`::
 
         sage: T20.category()
-        Category of finite dimensional modules over Algebra of differentiable
-         scalar fields on the 3-dimensional differentiable manifold R^3
+        Category of tensor products of finite dimensional modules over
+         Algebra of differentiable scalar fields on the 3-dimensional differentiable manifold R^3
         sage: T20.base_ring() is M.scalar_field_algebra()
         True
 
diff --git a/src/sage/manifolds/differentiable/vectorfield_module.py b/src/sage/manifolds/differentiable/vectorfield_module.py
index bd76a7547d6..580eec814ab 100644
--- a/src/sage/manifolds/differentiable/vectorfield_module.py
+++ b/src/sage/manifolds/differentiable/vectorfield_module.py
@@ -506,7 +506,7 @@ def destination_map(self):
         """
         return self._dest_map
 
-    def tensor_module(self, k, l):
+    def tensor_module(self, k, l, *, sym=None, antisym=None):
         r"""
         Return the module of type-`(k,l)` tensors on ``self``.
 
@@ -552,6 +552,8 @@ def tensor_module(self, k, l):
         for more examples and documentation.
 
         """
+        if sym or antisym:
+            raise NotImplementedError
         try:
             return self._tensor_modules[(k,l)]
         except KeyError:
@@ -1731,7 +1733,7 @@ def destination_map(self) -> DiffMap:
         """
         return self._dest_map
 
-    def tensor_module(self, k, l):
+    def tensor_module(self, k, l, *, sym=None, antisym=None):
         r"""
         Return the free module of all tensors of type `(k, l)` defined
         on ``self``.
@@ -1780,6 +1782,8 @@ def tensor_module(self, k, l):
             for more examples and documentation.
 
         """
+        if sym or antisym:
+            raise NotImplementedError
         try:
             return self._tensor_modules[(k,l)]
         except KeyError:
@@ -2031,7 +2035,7 @@ def basis(self, symbol=None, latex_symbol=None, from_frame=None,
                            symbol_dual=symbol_dual,
                            latex_symbol_dual=latex_symbol_dual)
 
-    def tensor(self, tensor_type, name=None, latex_name=None, sym=None,
+    def _tensor(self, tensor_type, name=None, latex_name=None, sym=None,
                antisym=None, specific_type=None):
         r"""
         Construct a tensor on ``self``.
diff --git a/src/sage/tensor/modules/finite_rank_free_module.py b/src/sage/tensor/modules/finite_rank_free_module.py
index 8c007727799..61a44d022d4 100644
--- a/src/sage/tensor/modules/finite_rank_free_module.py
+++ b/src/sage/tensor/modules/finite_rank_free_module.py
@@ -721,6 +721,28 @@ def map_isym(isym):
         result._index_maps = tuple(index_maps)
         return result
 
+    def tensor(self, *args, **kwds):
+        # Until https://trac.sagemath.org/ticket/30373 is done,
+        # TensorProductFunctor._functor_name is "tensor", so here we delegate.
+        r"""
+        Return the tensor product of ``self`` and ``others``.
+
+        This method is invoked when :class:`~sage.categories.tensor.TensorProductFunctor`
+        is applied to parents.
+
+        It just delegates to :meth:`tensor_product`.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(QQ, 2); M
+            2-dimensional vector space over the Rational Field
+            sage: M20 = M.tensor_module(2, 0); M20
+            Free module of type-(2,0) tensors on the 2-dimensional vector space over the Rational Field
+            sage: tensor([M20, M20])
+            Free module of type-(4,0) tensors on the 2-dimensional vector space over the Rational Field
+        """
+        return self.tensor_product(*args, **kwds)
+
     def rank(self) -> int:
         r"""
         Return the rank of the free module ``self``.
@@ -2122,7 +2144,7 @@ def _test_basis(self, tester=None, **options):
         TestSuite(b).run(verbose=tester._verbose, prefix=tester._prefix + "  ",
                          raise_on_failure=is_sub_testsuite)
 
-    def tensor(self, tensor_type, name=None, latex_name=None, sym=None,
+    def _tensor(self, tensor_type, name=None, latex_name=None, sym=None,
                antisym=None):
         r"""
         Construct a tensor on the free module ``self``.
@@ -2131,10 +2153,81 @@ def tensor(self, tensor_type, name=None, latex_name=None, sym=None,
 
         - ``tensor_type`` -- pair ``(k, l)`` with ``k`` being the
           contravariant rank and ``l`` the covariant rank
+
+        - ``name`` -- (default: ``None``) string; name given to the tensor
+
+        - ``latex_name`` -- (default: ``None``) string;  LaTeX symbol to
+          denote the tensor; if none is provided, the LaTeX symbol is set
+          to ``name``
+
+        - ``sym`` -- (default: ``None``) a symmetry or an iterable of symmetries
+          among the tensor arguments: each symmetry is described by a tuple
+          containing the positions of the involved arguments, with the
+          convention ``position = 0`` for the first argument. For instance:
+
+          * ``sym = (0,1)`` for a symmetry between the 1st and 2nd arguments
+          * ``sym = [(0,2), (1,3,4)]`` for a symmetry between the 1st and 3rd
+            arguments and a symmetry between the 2nd, 4th and 5th arguments.
+
+        - ``antisym`` -- (default: ``None``) antisymmetry or iterable of
+          antisymmetries among the arguments, with the same convention
+          as for ``sym``
+
+        OUTPUT:
+
+        - instance of
+          :class:`~sage.tensor.modules.free_module_tensor.FreeModuleTensor`
+          representing the tensor defined on ``self`` with the provided
+          characteristics
+
+        EXAMPLES:
+
+        Tensors on a rank-3 free module::
+
+            sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
+            sage: t = M._tensor((1,0), name='t') ; t
+            Element t of the Rank-3 free module M over the Integer Ring
+        """
+        from .comp import CompWithSym
+        sym, antisym = CompWithSym._canonicalize_sym_antisym(
+            tensor_type[0] + tensor_type[1], sym, antisym)
+        # Special cases:
+        if tensor_type == (1,0):
+            return self.element_class(self, name=name, latex_name=latex_name)
+        elif tensor_type == (0,1):
+            return self.linear_form(name=name, latex_name=latex_name)
+        elif tensor_type[0] == 0 and tensor_type[1] > 1 and antisym:
+            if len(antisym[0]) == tensor_type[1]:
+                return self.alternating_form(tensor_type[1], name=name,
+                                             latex_name=latex_name)
+        elif tensor_type[0] > 1 and tensor_type[1] == 0 and antisym:
+            if len(antisym[0]) == tensor_type[0]:
+                return self.alternating_contravariant_tensor(tensor_type[0],
+                                           name=name, latex_name=latex_name)
+        # Generic case:
+        return self.tensor_module(*tensor_type).element_class(self,
+                                 tensor_type, name=name, latex_name=latex_name,
+                                 sym=sym, antisym=antisym)
+
+    def tensor(self, *args, **kwds):
+        r"""
+        Construct a tensor on the free module ``self`` or a tensor product with other modules.
+
+        If ``args`` consist of other parents, just delegate to :meth:`tensor_product`.
+
+        Otherwise, construct a tensor from the following input.
+
+        INPUT:
+
+        - ``tensor_type`` -- pair ``(k, l)`` with ``k`` being the
+          contravariant rank and ``l`` the covariant rank
+
         - ``name`` -- (default: ``None``) string; name given to the tensor
+
         - ``latex_name`` -- (default: ``None``) string;  LaTeX symbol to
           denote the tensor; if none is provided, the LaTeX symbol is set
           to ``name``
+
         - ``sym`` -- (default: ``None``) a symmetry or an iterable of symmetries
           among the tensor arguments: each symmetry is described by a tuple
           containing the positions of the involved arguments, with the
@@ -2189,26 +2282,12 @@ def tensor(self, tensor_type, name=None, latex_name=None, sym=None,
             sage: M.tensor((3,0), antisym=[[]])
             Type-(3,0) tensor on the Rank-3 free module M over the Integer Ring
         """
-        from .comp import CompWithSym
-        sym, antisym = CompWithSym._canonicalize_sym_antisym(
-            tensor_type[0] + tensor_type[1], sym, antisym)
-        # Special cases:
-        if tensor_type == (1,0):
-            return self.element_class(self, name=name, latex_name=latex_name)
-        elif tensor_type == (0,1):
-            return self.linear_form(name=name, latex_name=latex_name)
-        elif tensor_type[0] == 0 and tensor_type[1] > 1 and antisym:
-            if len(antisym[0]) == tensor_type[1]:
-                return self.alternating_form(tensor_type[1], name=name,
-                                             latex_name=latex_name)
-        elif tensor_type[0] > 1 and tensor_type[1] == 0 and antisym:
-            if len(antisym[0]) == tensor_type[0]:
-                return self.alternating_contravariant_tensor(tensor_type[0],
-                                           name=name, latex_name=latex_name)
-        # Generic case:
-        return self.tensor_module(*tensor_type).element_class(self,
-                                 tensor_type, name=name, latex_name=latex_name,
-                                 sym=sym, antisym=antisym)
+        # Until https://trac.sagemath.org/ticket/30373 is done,
+        # TensorProductFunctor._functor_name is "tensor", so this method
+        # also needs to double as the tensor product construction
+        if isinstance(args[0], Parent):
+            return self.tensor_product(*args, **kwds)
+        return self._tensor(*args, **kwds)
 
     def tensor_from_comp(self, tensor_type, comp, name=None, latex_name=None):
         r"""
diff --git a/src/sage/tensor/modules/tensor_free_module.py b/src/sage/tensor/modules/tensor_free_module.py
index 39174258ebb..b04e8581148 100644
--- a/src/sage/tensor/modules/tensor_free_module.py
+++ b/src/sage/tensor/modules/tensor_free_module.py
@@ -58,6 +58,7 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
+from sage.categories.modules import Modules
 from sage.misc.cachefunc import cached_method
 from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule_abstract
 from sage.tensor.modules.free_module_tensor import FreeModuleTensor
@@ -126,7 +127,7 @@ class TensorFreeModule(FiniteRankFreeModule_abstract):
     ``T`` is a module (actually a free module) over `\ZZ`::
 
         sage: T.category()
-        Category of finite dimensional modules over Integer Ring
+        Category of tensor products of finite dimensional modules over Integer Ring
         sage: T in Modules(ZZ)
         True
         sage: T.rank()
@@ -336,7 +337,7 @@ class TensorFreeModule(FiniteRankFreeModule_abstract):
 
     Element = FreeModuleTensor
 
-    def __init__(self, fmodule, tensor_type, name=None, latex_name=None):
+    def __init__(self, fmodule, tensor_type, name=None, latex_name=None, category=None):
         r"""
         TESTS::
 
@@ -347,33 +348,47 @@ def __init__(self, fmodule, tensor_type, name=None, latex_name=None):
         """
         self._fmodule = fmodule
         self._tensor_type = tuple(tensor_type)
+        ring = fmodule._ring
         rank = pow(fmodule._rank, tensor_type[0] + tensor_type[1])
         if self._tensor_type == (0,1):  # case of the dual
+            category = Modules(ring).FiniteDimensional().or_subcategory(category)
             if name is None and fmodule._name is not None:
                 name = fmodule._name + '*'
             if latex_name is None and fmodule._latex_name is not None:
                 latex_name = fmodule._latex_name + r'^*'
         else:
+            category = Modules(ring).FiniteDimensional().TensorProducts().or_subcategory(category)
             if name is None and fmodule._name is not None:
                 name = 'T^' + str(self._tensor_type) + '(' + fmodule._name + \
                        ')'
             if latex_name is None and fmodule._latex_name is not None:
                 latex_name = r'T^{' + str(self._tensor_type) + r'}\left(' + \
                              fmodule._latex_name + r'\right)'
-        super().__init__(fmodule._ring, rank, name=name, latex_name=latex_name)
+        super().__init__(fmodule._ring, rank, name=name, latex_name=latex_name, category=category)
         fmodule._all_modules.add(self)
 
-    def construction(self):
+    def tensor_factors(self):
         r"""
-        TESTS::
+        Return the tensor factors of this tensor module.
+
+        EXAMPLES::
 
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
             sage: T = M.tensor_module(2, 3)
-            sage: T.construction() is None
-            True
+            sage: T.tensor_factors()
+            [Rank-3 free module M over the Integer Ring,
+             Rank-3 free module M over the Integer Ring,
+             Dual of the Rank-3 free module M over the Integer Ring,
+             Dual of the Rank-3 free module M over the Integer Ring,
+             Dual of the Rank-3 free module M over the Integer Ring]
         """
-        # No construction until https://trac.sagemath.org/ticket/31276 provides tensor_product methods
-        return None
+        if self._tensor_type == (0,1):  # case of the dual
+            raise NotImplementedError
+        factors = [self._fmodule] * self._tensor_type[0]
+        dmodule = self._fmodule.dual()
+        if self._tensor_type[1]:
+            factors += [dmodule] * self._tensor_type[1]
+        return factors
 
     #### Parent Methods
 
diff --git a/src/sage/tensor/modules/tensor_free_submodule.py b/src/sage/tensor/modules/tensor_free_submodule.py
index 5f6964f8546..ff4f739ac0d 100644
--- a/src/sage/tensor/modules/tensor_free_submodule.py
+++ b/src/sage/tensor/modules/tensor_free_submodule.py
@@ -178,6 +178,24 @@ def power_name(op, s, latex=False):
                                                latex_name=latex_name,
                                                category=category, ambient=ambient)
 
+    def construction(self):
+        # TODO: Define the symmetry group and its action (https://trac.sagemath.org/ticket/34495),
+        # return the construction functor for invariant subobjects.
+        r"""
+        Return the functorial construction of ``self``.
+
+        This implementation just returns ``None``.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
+            sage: Sym2M = M.tensor_module(2, 0, sym=range(2)); Sym2M
+            Free module of fully symmetric type-(2,0) tensors on the Rank-3 free module M over the Integer Ring
+            sage: Sym2M.construction() is None
+            True
+        """
+        return None
+
     @cached_method
     def _basis_sym(self):
         r"""