Lemma commutative_symmetric_braiding_using_lrwhisker (m : commutative_comonoid M) : δ_{m} · (m ⊗^{M}_{l} δ_{m}) · (m ⊗^{M}_{l} (m ⊗^{M}_{l} δ_{m})) · m ⊗^{M}_{l} mon_rassociator _ _ _ = δ_{m} · (m ⊗^{M}_{l} δ_{m}) · (m ⊗^{M}_{l} (δ_{m} ⊗^{M}_{r} m)). Proof. rewrite ! assoc'. apply maponpaths. rewrite <- (bifunctor_leftcomp M). etrans. { apply pathsinv0, (bifunctor_leftcomp M). } etrans. 2: { apply (bifunctor_leftcomp M). } apply maponpaths. refine (_ @ ! comonoid_laws_assoc' M m). apply assoc. 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.