Lemma dialgebra_disp_leftunitor_data : disp_leftunitor_data dialgebra_disp_tensor dialgebra_disp_unit. Proof. intros a f. cbn. unfold dialgebra_disp_unit, dialgebra_disp_tensor_op. rewrite (bifunctor_equalwhiskers W). unfold functoronmorphisms2. rewrite bifunctor_rightcomp. set (aux1 := fmonoidal_preservesleftunitality Gm a). etrans. { repeat rewrite assoc'. do 3 apply maponpaths. rewrite assoc. exact aux1. } clear aux1. apply (z_iso_inv_on_right _ _ _ (_,,fmonoidal_preservestensorstrongly Fm I_{V} a)). cbn. set (aux2 := fmonoidal_preservesleftunitality Fm a). etrans. { rewrite assoc. apply cancel_postcomposition. apply pathsinv0, (bifunctor_equalwhiskers W). } unfold functoronmorphisms1. rewrite assoc'. etrans. { apply maponpaths. apply monoidal_leftunitornat. } repeat rewrite assoc. apply cancel_postcomposition. rewrite <- aux2. clear aux2. repeat rewrite assoc. apply cancel_postcomposition. rewrite <- (bifunctor_rightcomp W). etrans. { apply cancel_postcomposition. apply maponpaths. apply (z_iso_after_z_iso_inv (_,,fmonoidal_preservesunitstrongly Fm)). } rewrite bifunctor_rightid. apply id_left. 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.Categories.Dialgebras. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. 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.Displayed.MonoidalSections. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Displayed.MonoidalFunctorLifting. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import BifunctorNotations. Import MonoidalNotations. Import DisplayedBifunctorNotations. Import DisplayedMonoidalNotations. Section FromMonoidalSection. End FromMonoidalSection.