Lemma id_to_teqi'''_to_id {C : bicat_of_univ_cats} {TC TD : bidisp_tensor_disp_bicat C} (p : TC = TD) : tensor_eqi'''_to_eq TC TD (id_tensor_eqi''' TC TD p) = p. Proof. unfold tensor_eqi'''_to_eq. unfold id_tensor_eqi'''. induction p. use pathscomp0. 3: apply total2_paths_idpath. apply maponpaths_total. - use pathscomp0. 3: apply total2_paths_idpath. apply maponpaths_total. + apply funextsec_id. + intro T. repeat (apply impred_isaset ; intro). apply homset_property. - intro T. apply isaset_dirprod; repeat (apply impred_isaset ; intro) ; apply isasetaprop ; apply homset_property. 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.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.CurriedMonoidalCategories. Import TensorNotation. Import TensorNotation.