Definition isaprop_global_cleaving {B : bicat} (HB : is_univalent_2 B) (D : disp_bicat B) (HD : disp_univalent_2 D) : isaprop (global_cleaving D). Proof. use impred ; intro x. use impred ; intro y. use impred ; intro xx. use impred ; intro f. use invproofirrelevance. intros ℓ₁ ℓ₂. use eq_cartesian_lift_from_adjequiv. - apply HB. - apply HD. - refine (map_between_cartesian_1cell (pr2 HB) (pr12 ℓ₁) (pr22 ℓ₂) ,, _). refine (transportf (λ z, disp_left_adjoint_equivalence z _) _ (pr2 (disp_adj_equiv_between_cartesian_1cell (pr2 HB) (pr22 ℓ₁) (pr22 ℓ₂)))). use unique_internal_adjoint_equivalence. apply HB. - cbn. apply map_between_cartesian_1cell_commute. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Opposite. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat. Require Import UniMath.Bicategories.DisplayedBicats.Cartesians. Require Import UniMath.Bicategories.DisplayedBicats.EquivalenceBetweenCartesians.