proposition: Lemma sigma_disp_is_tensor : is_disp_bifunctor M sigma_disp_cat_tensor_data.
proof: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Projection UniMath.CategoryTheory.DisplayedCats.Examples.Sigma UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Displayed.Symmetric UniMath.CategoryTheory.Monoidal.Examples.Fullsub UniMath.CategoryTheory.Monoidal.Structure.Symmetric