Lemma comonoid_disp_lunitor (m : comonoid M) : comonoid_mor_struct M (tensor_of_comonoids comonoid_disp_unit m) m (mon_lunitor m). Proof. use make_is_comonoid_mor. - cbn. etrans. { apply maponpaths. apply pathsinv0. apply (precompose_inner_swap_with_lunitors_on_right (M)). } refine (_ @ monoidal_leftunitornat M _ _ δ_{m}). rewrite ! assoc. apply maponpaths_2. etrans. { apply maponpaths_2. rewrite assoc'. apply maponpaths. apply inner_swap_inv. } rewrite id_right. rewrite <- (when_bifunctor_becomes_rightwhiskering M). etrans. { apply pathsinv0, tensor_comp_mor. } rewrite id_right. rewrite <- (when_bifunctor_becomes_leftwhiskering M). apply maponpaths_2. apply (monoidal_leftunitorisolaw M I_{M}). - refine (_ @ monoidal_leftunitornat M _ _ ε_{m}). rewrite id_right. cbn. apply maponpaths_2. apply (when_bifunctor_becomes_leftwhiskering 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.