Definition intersection_with_full_is_contravariant {B : bicat_with_pb} (P₀ : ∏ (x y : B), x --> y → UU) (closed_pb : ∏ (pb x y z : B) (f : x --> z) (g : y --> z) (p₁ : pb --> x) (p₂ : pb --> y) (γ : invertible_2cell (p₁ · f) (p₂ · g)), P₀ x z f → has_pb_ump (make_pb_cone pb p₁ p₂ γ) → P₀ pb y p₂) {D : disp_map_bicat B} (HD : is_contravariant_disp_map_bicat D) : is_contravariant_disp_map_bicat (intersection_disp_map_bicat D (full_disp_map_bicat P₀ closed_pb)). Proof. split. - intros x y f Hf. apply HD. apply Hf. - intros e₁ e₂ b₁ b₂ p₁ p₂ fe. split. + intro H. apply HD. apply H. + intro H. split. * apply HD. apply H. * exact tt. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. 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.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Morphisms.Properties.ContainsAdjEquiv. Require Import UniMath.Bicategories.Morphisms.Properties.Composition. Require Import UniMath.Bicategories.Morphisms.Properties.ClosedUnderPullback. Require Import UniMath.Bicategories.Morphisms.Properties.FromInitial. Require Import UniMath.Bicategories.Colimits.Initial. Require Import UniMath.Bicategories.Limits.Pullbacks.