proposition: Lemma tensor_of_comm_comonoids (mx my : commutative_comonoid V) : is_commutative V (tensor_of_comonoids V mx my).
proof: Proof. use (z_iso_inv_on_left _ _ _ _ (inner_swap V _ _ _ _ ,, inner_swap V _ _ _ _ ,, _)). { apply inner_swap_is_z_isomorphism. } cbn. rewrite assoc'. etrans. 2: { apply maponpaths. apply inner_swap_commute_with_swap. } rewrite assoc. etrans. 2: { apply maponpaths_2. rewrite assoc'. apply maponpaths. apply pathsinv0, inner_swap_inv. } rewrite id_right. refine (_@ tensor_comp_mor _ _ _ _). etrans. 2: { apply maponpaths_2. apply pathsinv0, commutative_comonoid_is_commutative. } apply maponpaths. apply pathsinv0, commutative_comonoid_is_commutative. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Examples.Fullsub UniMath.CategoryTheory.Monoidal.Examples.Sigma UniMath.CategoryTheory.Monoidal.Structure.Cartesian UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal UniMath.CategoryTheory.Monoidal.Displayed.Symmetric UniMath.CategoryTheory.Monoidal.Comonoids.Category