Lemma bidisp_ass_disp_prebicat_is_globally_univalent : disp_univalent_2_0 bidisp_ass_disp_bicat. Proof. apply disp_cell_unit_bicat_univalent_2_0. - apply bidisp_tensorunit_is_univalent_2. - intro ; intros. apply isaprop_preserves_associator. - intro ; apply isaset_associator. - intros C a1 a2 pa. use total2_paths_f. + apply funextsec ; intro x1 ; apply funextsec ; intro x2 ; apply funextsec ; intro x3. set (p := pr1 pa x1 x2 x3). cbn in p. unfold identityfunctor_preserves_tensor_data in p. rewrite id_right in p. rewrite id_right in p. rewrite tensor_id in p. rewrite id_left in p. rewrite tensor_id in p. rewrite id_right in p. exact p. + apply isaprop_associator_nat. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. 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.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sigma. Require Import  UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.CurriedMonoidalCategories. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.TensorLayer. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.UnitLayer. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.TensorUnitLayer.