Definition local_cleaving_of_disp_map_bicat_cartesian : pred_mor D (pr12 xx) (pr12 yy) ff. Proof. apply HD. intros z h₁ h₂ δ Hδ. assert (H₁ : is_cartesian_2cell_sfib (pr12 yy) (h₂ ◃ β)). { apply (pr2 H). apply internal_sfib_cleaving_is_cartesian. } assert (H₂ : is_cartesian_2cell_sfib (pr12 yy) (h₁ ◃ β)). { apply (pr2 H). apply internal_sfib_cleaving_is_cartesian. } assert (H₃ : is_cartesian_2cell_sfib (pr12 yy) (δ ▹ pr1 gg)). { apply Hgg. exact Hδ. } use (is_cartesian_2cell_sfib_postcomp _ _ H₁ (vcomp_is_cartesian_2cell_sfib _ H₂ H₃)). abstract (rewrite vcomp_whisker ; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.StreetFibration. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Logic.DisplayMapBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayMapBicatToDispBicat. Require Import UniMath.Bicategories.Limits.Pullbacks.