Lemma sigma_disp_is_tensor : is_disp_bifunctor M sigma_disp_cat_tensor_data. Proof. repeat split. - intros x y [xx xxx] [yy yyy]. use total2_paths_f. + refine (disp_bifunctor_leftid DM x y xx yy @ _). apply pathsinv0, pr1_transportf. + apply E_prop. - intros x y [xx xxx] [yy yyy]. use total2_paths_f. + refine (disp_bifunctor_rightid DM x y xx yy @ _). apply pathsinv0, pr1_transportf. + apply E_prop. - intro ; intros. use total2_paths_f. + refine (disp_bifunctor_leftcomp DM _ _ _ _ _ _ _ _ _ _ _ _ @ _). apply pathsinv0, pr1_transportf. + apply E_prop. - intro ; intros. use total2_paths_f. + refine (disp_bifunctor_rightcomp DM _ _ _ _ _ _ _ _ _ _ _ _ @ _). apply pathsinv0, pr1_transportf. + apply E_prop. - intro ; intros. use total2_paths_f. + etrans. { apply disp_bifunctor_equalwhiskers. } apply pathsinv0, pr1_transportf. + apply E_prop. 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.