Proposition make_commutative_comonoid_laws : comonoid_laws V make_commutative_comonoid_data. Proof. repeat split. - refine (_ @ unit_left) ; cbn. apply maponpaths_2. apply maponpaths. rewrite tensor_mor_right. apply idpath. - refine (_ @ unit_left) ; cbn. refine (!_). etrans. { do 2 apply maponpaths_2. exact (!(comm)). } rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_sym_mon_braiding. rewrite !assoc'. rewrite sym_mon_braiding_lunitor. apply maponpaths_2. rewrite tensor_mor_left. apply idpath. - unfold comonoid_laws_assoc ; cbn. refine (_ @ assocδ @ _). + apply maponpaths_2. apply maponpaths. rewrite tensor_mor_right. apply idpath. + apply maponpaths. rewrite tensor_mor_left. apply idpath. 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.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Projection. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Sigma. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Import BifunctorNotations. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal. Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian. 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.Displayed.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Examples.Sigma. Require Import UniMath.CategoryTheory.Monoidal.Examples.Fullsub. Require Import UniMath.CategoryTheory.Monoidal.Examples.DiagonalFunctor. Require Import UniMath.CategoryTheory.Monoidal.Examples.ConstantFunctor. Require Import UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids. Require Import UniMath.CategoryTheory.Categories.Dialgebras. Import MonoidalNotations. Import ComonoidNotations. Import ComonoidNotations. Import ComonoidNotations. Import ComonoidNotations. Import ComonoidNotations.