Theorem bidisp_tensor_disp_bicat_is_locally_univalent : disp_univalent_2_1 bidisp_tensor_disp_bicat. Proof. apply fiberwise_local_univalent_is_univalent_2_1. intros C D F TC TD ptF1 ptF2. use isweqimplimpl. - apply disp_inv_to_eq. - cbn in ptF1 ; cbn in ptF2. rewrite idpath_transportf. unfold preserves_tensor in ptF1 ; unfold preserves_tensor in ptF2. assert (pf : isaset (preserves_tensor TC TD F)). { apply isaset_total2. - repeat (apply impred_isaset ; intro). apply univalent_category_has_homsets. - intro pt. apply isasetaprop. repeat (apply impred_isaprop ; intro). apply univalent_category_has_homsets. } apply pf. - apply isaproptotal2. + intro. apply isaprop_is_disp_invertible_2cell. + intros. apply isaprop_bidisp_tensor_disp_2cell_struct. Defined.
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.