Lemma precompose_inner_swap_with_augs_on_left (mx my : comonoid M) : inner_swap M mx mx my my · (ε_{ mx} ⊗^{ M} ε_{ my}) ⊗^{ M}_{r} (mx ⊗_{ M} my) = (ε_{ mx} ⊗^{M}_{r} _) #⊗ (ε_{ my} ⊗^{M}_{r} my) · inner_swap M _ _ _ _. Proof. refine (_ @ naturality_inner_swap (M) ε_{mx} (identity _) ε_{my} (identity _) @ _). - rewrite <- (when_bifunctor_becomes_rightwhiskering M). do 2 apply maponpaths. apply pathsinv0, tensor_id_id. - now rewrite <- ! (when_bifunctor_becomes_rightwhiskering M). 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.