Lemma tensor_of_comm_comonoids (mx my : commutative_comonoid V) : is_commutative V (tensor_of_comonoids V mx my). 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.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Import BifunctorNotations. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Monoidal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.CategoryTheory.Monoidal.Examples.Fullsub. Require Import UniMath.CategoryTheory.Monoidal.Examples.Sigma. Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Category. Import MonoidalNotations. Import ComonoidNotations.