Definition intersection_arrow_subbicat {B : bicat} (D₁ D₂ : arrow_subbicat B) : arrow_subbicat B. Proof. use make_arrow_subbicat. - exact (λ x y f, pred_ob D₁ f × pred_ob D₂ f). - exact (λ _ _ _ _ p₁ p₂ fe, pred_mor D₁ p₁ p₂ fe × pred_mor D₂ p₁ p₂ fe). - intro ; intros. split ; apply id_pred_mor. - intros ? ? ? ? ? ? ? ? ? ? ? H₁ H₂. split. + exact (comp_pred_mor D₁ (pr1 H₁) (pr1 H₂)). + exact (comp_pred_mor D₂ (pr2 H₁) (pr2 H₂)). - intros e₁ e₂ b₁ b₂ p₁ p₁' p₂ p₂' fe fe' α β γ H. split. + exact (invertible_pred_mor D₁ α β γ (pr1 H)). + exact (invertible_pred_mor D₂ α β γ (pr2 H)). 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.