Lemma tensor_of_comonoids_laws_unit_left (mx my : comonoid M) : comonoid_laws_unit_left M (tensor_of_comonoids_data mx my). Proof. unfold comonoid_laws_unit_left. cbn. rewrite (bifunctor_rightcomp M). rewrite ! assoc. etrans. { do 2 apply maponpaths_2. rewrite assoc'. apply maponpaths. apply precompose_inner_swap_with_augs_on_left. } rewrite assoc. etrans. { do 3 apply maponpaths_2. apply pathsinv0, tensor_comp_mor. } etrans. { rewrite ! assoc'. apply maponpaths. rewrite assoc. exact (precompose_inner_swap_with_lunitors_on_right (M) mx my). } cbn. etrans. { apply pathsinv0. apply tensor_comp_mor. } etrans. { apply maponpaths. apply comonoid_to_law_unit_left. } etrans. { apply maponpaths_2. apply comonoid_to_law_unit_left. } 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.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.