Lemma diagonal_is_comonoid_mor_counit (m : comonoid V) : is_comonoid_mor_counit V m (tensor_of_comonoids V m m) δ_{m}. Proof. unfold is_comonoid_mor_counit. cbn. unfold monoidal_cat_tensor_mor. unfold functoronmorphisms1. rewrite ! assoc'. etrans. { apply maponpaths. apply maponpaths. exact (monoidal_leftunitornat V _ _ ε_{m}). } refine (_ @ id_left _). rewrite ! assoc. apply maponpaths_2. exact (comonoid_to_law_unit_left _ 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.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.Displayed.Symmetric. Require Import UniMath.CategoryTheory.Categories.Dialgebras. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalDialgebras. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Category. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Tensor. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Monoidal. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.MonoidalCartesianBuilder. Import MonoidalNotations. Import ComonoidNotations.