Lemma whisker_to_total' (x y : V) : mon_runitor x #⊗ mon_lunitor y · δ_{cm (x ⊗ y)} · (x ⊗^{V}_{l} ε_{cm y}) #⊗ (ε_{cm x} ⊗^{V}_{r} y) = identity _. Proof. rewrite diagonal_of_tensor. etrans. { rewrite ! assoc'. do 2 apply maponpaths. apply inner_swap_before_aug. } etrans. { apply maponpaths. apply pathsinv0, tensor_comp_mor. } refine (! tensor_comp_mor _ _ _ _ @ _). rewrite comonoid_unit_law_left_inv. rewrite comonoid_unit_law_right_inv. apply tensor_id_id. 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.