Definition intersection_arrow_subbicat_composition {B : bicat} (D₁ D₂ : arrow_subbicat B) (HD₁ : arrow_subbicat_closed_composition D₁) (HD₂ : arrow_subbicat_closed_composition D₂) : arrow_subbicat_closed_composition (intersection_arrow_subbicat D₁ D₂). Proof. split. - intros x y z f g Hf Hg. split. + apply HD₁. * exact (pr1 Hf). * exact (pr1 Hg). + apply HD₂. * exact (pr2 Hf). * exact (pr2 Hg). - intros x y z f g Hf Hg. split. + apply HD₁. * exact (pr1 Hf). * exact (pr1 Hg). + apply HD₂. * exact (pr2 Hf). * exact (pr2 Hg). 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.