Lemma dialgebra_disp_tensor_comp_aux2 {a1 a1' a : A} {h: a1 --> a1'} (f : base_disp a1) (f' : base_disp a) (g : base_disp a1') : f -->[h] g -> dialgebra_disp_tensor_op f f' -->[h ⊗^{V}_{r} a] dialgebra_disp_tensor_op g f'. Proof. intro Hyp. cbn in Hyp. cbn. unfold dialgebra_disp_tensor_op. etrans. { repeat rewrite assoc'. do 2 apply maponpaths. apply pathsinv0, fmonoidal_preservestensornatright. } repeat rewrite assoc. apply cancel_postcomposition. etrans. { unfold functoronmorphisms1. repeat rewrite assoc'. do 2 apply maponpaths. apply pathsinv0, (bifunctor_equalwhiskers W). } unfold functoronmorphisms1. repeat rewrite assoc. apply cancel_postcomposition. assert (aux := fmonoidal_preservestensornatright Fm _ _ a h). apply (z_iso_inv_on_right _ _ _ (_,,fmonoidal_preservestensorstrongly Fm a1 a)) in aux. rewrite assoc in aux. apply pathsinv0 in aux. apply (z_iso_inv_on_left _ _ _ _ (_,,fmonoidal_preservestensorstrongly Fm a1' a)) in aux. etrans. 2: { apply cancel_postcomposition. apply aux. } clear aux. do 2 rewrite assoc'. apply maponpaths. do 2 rewrite <- (bifunctor_rightcomp W). apply maponpaths. exact Hyp. 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.