Lemma comonoid_disp_braiding (mx my : comonoid M) : comonoid_mor_struct M (tensor_of_comonoids mx my) (tensor_of_comonoids my mx) (sym_mon_braiding M mx my). Proof. apply make_is_comonoid_mor. - cbn. etrans. 2: { rewrite assoc. apply maponpaths_2. apply (tensor_sym_mon_braiding). } rewrite ! assoc'. apply comult_before_inner_swap_and_swap. - cbn. rewrite ! assoc. rewrite id_right. apply maponpaths_2. etrans. 2: { apply (tensor_sym_mon_braiding M). } cbn. refine (! id_right _ @ _). apply maponpaths. apply pathsinv0, sym_mon_braiding_id. 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.