Lemma sigma_disp_cat_monoidal_braiding_laws : disp_braiding_laws (sigma_disp_cat_monoidal EM E_prop) sigma_disp_cat_monoidal_braiding_data sigma_disp_cat_monoidal_braiding_data. Proof. repeat split ; try (intro ; intros) ; use total2_paths_f ; try (apply E_prop). - etrans. { apply pr1_transportf. } apply (disp_braiding_to_naturality_left DM DS). - etrans. { apply pr1_transportf. } apply (disp_braiding_to_naturality_right DM DS). - etrans. { apply (disp_braiding_to_inverses DM DS). } apply pathsinv0, pr1_transportf. - etrans. { apply (disp_braiding_to_inverses DM DS). } etrans. 2: { apply pathsinv0, pr1_transportf. } unfold transportb. apply maponpaths_2. apply homset_property. - etrans. 2: { apply (disp_braiding_to_hexagon1 _ DS). } apply pr1_transportf. - etrans. 2: { apply (disp_braiding_to_hexagon2 _ DS). } apply pr1_transportf. 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.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Projection. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Sigma. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Import BifunctorNotations. 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.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Examples.Fullsub. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import MonoidalNotations. Import DisplayedBifunctorNotations. Import DisplayedMonoidalNotations.