Lemma dialgebra_disp_symmetric_data : disp_symmetric_data base_mon_disp HV. Proof. intros x y xx yy. red in xx, yy. cbn in xx, yy. cbn. unfold dialgebra_disp_tensor_op. repeat rewrite assoc'. apply (z_iso_inv_on_right _ _ _ (_,,fmonoidal_preservestensorstrongly Fm x y)). etrans. { apply maponpaths. apply pathsinv0, Gs. } repeat rewrite assoc. apply cancel_postcomposition. cbn. etrans. 2: { do 2 apply cancel_postcomposition. exact (Fs x y). } etrans. 2: { apply cancel_postcomposition. rewrite assoc'. apply maponpaths. apply pathsinv0, (z_iso_inv_after_z_iso (_ ,, fmonoidal_preservestensorstrongly Fm y x)). } rewrite id_right. apply (tensor_sym_mon_braiding (((B,,W):monoidal_cat),,HW)). 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.Structure.Symmetric. 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.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalDialgebras. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Import BifunctorNotations. Import MonoidalNotations. Import DisplayedBifunctorNotations. Import DisplayedMonoidalNotations.