Lemma dialgebra_disp_tensor_comp_aux1 {a a2 a2' : A} {h': a2 --> a2'} (f : base_disp a) (f' : base_disp a2) (g' : base_disp a2') : f'-->[h'] g' -> dialgebra_disp_tensor_op f f' -->[a ⊗^{V}_{l} h'] dialgebra_disp_tensor_op f g'. Proof. intro Hyp'. cbn in Hyp'. cbn. unfold dialgebra_disp_tensor_op. etrans. { repeat rewrite assoc'. do 2 apply maponpaths. apply pathsinv0, fmonoidal_preservestensornatleft. } repeat rewrite assoc. apply cancel_postcomposition. etrans. { rewrite (bifunctor_equalwhiskers W). unfold functoronmorphisms2. repeat rewrite assoc'. do 2 apply maponpaths. apply (bifunctor_equalwhiskers W). } rewrite (bifunctor_equalwhiskers W). unfold functoronmorphisms2. repeat rewrite assoc. apply cancel_postcomposition. assert (aux := fmonoidal_preservestensornatleft Fm a a2 a2' h'). apply (z_iso_inv_on_right _ _ _ (_,,fmonoidal_preservestensorstrongly Fm a a2)) in aux. rewrite assoc in aux. apply pathsinv0 in aux. apply (z_iso_inv_on_left _ _ _ _ (_,,fmonoidal_preservestensorstrongly Fm a a2')) in aux. etrans. 2: { apply cancel_postcomposition. apply aux. } clear aux. do 2 rewrite assoc'. apply maponpaths. do 2 rewrite <- (bifunctor_leftcomp 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.