proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Projection UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Examples.Sigma UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Structure.Symmetric UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal UniMath.CategoryTheory.Monoidal.Structure.Cartesian UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Displayed.Symmetric UniMath.CategoryTheory.Monoidal.Examples.Sigma UniMath.CategoryTheory.Monoidal.Examples.Fullsub UniMath.CategoryTheory.Monoidal.Examples.DiagonalFunctor UniMath.CategoryTheory.Monoidal.Examples.ConstantFunctor UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids UniMath.CategoryTheory.Categories.Dialgebras