Lemma make_disp_symm_braiding_nat_right {B : symmetric M} {c : disp_braiding_data DM B} (p_nat : disp_sym_moncat_laws_tensored_nat c) : disp_braiding_law_naturality_right DM c. Proof. intro ; intros. set (q := ! p_nat _ _ _ _ _ _ xx1 xx2 yy yy ff (id_disp yy)). use transportf_transpose_left. set (qq := specialized_transport_lemma (c x1 y xx1 yy) ff _ _ q). refine (qq @ _). clear qq. clear q. use transportf_transpose_left. unfold transportb. rewrite transport_f_f. rewrite path_comp_inv_inv. etrans. 2: { apply maponpaths_2. apply maponpaths. refine (idpath (maponpaths (λ i, i · pr1 B _ _) (when_bifunctor_becomes_rightwhiskering M y f)) @ _). apply homset_property. } etrans. 2: { apply maponpaths_2. apply (maponpathsinv0 (λ i : C ⟦ x1 ⊗_{ M} y, x2 ⊗_{ M} y ⟧, i · pr1 B x2 y)). } etrans. 2: { apply mor_disp_transportf_postwhisker. } apply maponpaths_2. apply disp_tensor_with_id_on_right. 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.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Projection. 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.TransportLemmas. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import BifunctorNotations. Import MonoidalNotations. Import DisplayedBifunctorNotations. Import DisplayedMonoidalNotations.