gh-36399: fixing category of ncsym and dual
    
by making them Commutative and CoCommutative resp.

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
    
URL: #36399
Reported by: Frédéric Chapoton
Reviewer(s): Darij Grinberg

Author: Release Manager

src/sage/combinat/ncsym/dual.py

@@ -6,12 +6,12 @@
 
 - Travis Scrimshaw (08-04-2013): Initial version
 """
-#*****************************************************************************
+# ****************************************************************************
 #       Copyright (C) 2013 Travis Scrimshaw <tscrim at ucdavis.edu>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
-#                  http://www.gnu.org/licenses/
-#*****************************************************************************
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
 
 from sage.misc.lazy_attribute import lazy_attribute
 from sage.misc.misc_c import prod
@@ -49,9 +49,9 @@ def __init__(self, R):
             sage: TestSuite(SymmetricFunctionsNonCommutingVariables(QQ).dual()).run()
         """
         # change the line below to assert R in Rings() once MRO issues from #15536, #15475 are resolved
-        assert R in Fields() or R in Rings() # side effect of this statement assures MRO exists for R
+        assert R in Fields() or R in Rings()  # side effect of this statement assures MRO exists for R
         self._base = R  # Won't be needed once CategoryObject won't override base_ring
-        category = GradedHopfAlgebras(R)  # TODO: .Commutative()
+        category = GradedHopfAlgebras(R).Commutative()
         Parent.__init__(self, category=category.WithRealizations())
 
         # Bases
@@ -104,7 +104,7 @@ def dual(self):
 
     class w(NCSymBasis_abstract):
         r"""
-        The Hopf algebra of symmetric functions in non-commuting variables
+        The dual Hopf algebra of symmetric functions in non-commuting variables
         in the `\mathbf{w}` basis.
 
         EXAMPLES::
@@ -135,10 +135,14 @@ def __init__(self, NCSymD):
             """
             def key_func_set_part(A):
                 return sorted(map(sorted, A))
-            CombinatorialFreeModule.__init__(self, NCSymD.base_ring(), SetPartitions(),
+
+            R = NCSymD.base_ring()
+            category = GradedHopfAlgebras(R).Commutative()
+            category &= NCSymDualBases(NCSymD)
+            CombinatorialFreeModule.__init__(self, R, SetPartitions(),
                                              prefix='w', bracket=False,
                                              sorting_key=key_func_set_part,
-                                             category=NCSymDualBases(NCSymD))
+                                             category=category)
 
         @lazy_attribute
         def to_symmetric_function(self):

src/sage/combinat/ncsym/ncsym.py

@@ -297,7 +297,7 @@ def __init__(self, R):
         # change the line below to assert R in Rings() once MRO issues from #15536, #15475 are resolved
         assert R in Fields() or R in Rings()  # side effect of this statement assures MRO exists for R
         self._base = R  # Won't be needed once CategoryObject won't override base_ring
-        category = GradedHopfAlgebras(R)  # TODO: .Cocommutative()
+        category = GradedHopfAlgebras(R).Cocommutative()
         Parent.__init__(self, category=category.WithRealizations())
 
     def _repr_(self):
@@ -362,9 +362,13 @@ def __init__(self, NCSym):
                 sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
                 sage: TestSuite(NCSym.m()).run()
             """
-            CombinatorialFreeModule.__init__(self, NCSym.base_ring(), SetPartitions(),
+            R = NCSym.base_ring()
+            category = GradedHopfAlgebras(R).Cocommutative()
+            category &= NCSymBases(NCSym)
+
+            CombinatorialFreeModule.__init__(self, R, SetPartitions(),
                                              prefix='m', bracket=False,
-                                             category=NCSymBases(NCSym))
+                                             category=category)
 
         @cached_method
         def _m_to_p_on_basis(self, A):