Lemma inner_swap_before_aug (x y : V) : inner_swap V x x y y · (x ⊗^{V}_{l} ε_{cm y}) #⊗ (ε_{cm _} ⊗^{V}_{r} y) = (_ ⊗^{V}_{l} ε_{cm _}) #⊗ (ε_{cm _} ⊗^{V}_{r} _). Proof. refine (_ @ naturality_inner_swap V (identity x) ε_{cm x} ε_{cm y} (identity y) @ _). { now rewrite <- (when_bifunctor_becomes_leftwhiskering V), <- (when_bifunctor_becomes_rightwhiskering V). } rewrite <- (when_bifunctor_becomes_leftwhiskering V), <- (when_bifunctor_becomes_rightwhiskering V). rewrite inner_swap_along_unit. apply id_right. 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.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.Monoidal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. 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.Comonoids.Category. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Tensor. Import MonoidalNotations. Import ComonoidNotations. Import ComonoidNotations.