Definition sfib_subbicat (B : bicat) : arrow_subbicat B. Proof. use make_arrow_subbicat. - exact (λ _ _ f, internal_sfib f). - exact (λ _ _ _ _ p₁ p₂ fe, mor_preserves_cartesian p₁ p₂ fe). - intro ; intros. apply id_mor_preserves_cartesian. - intros e₁ e₂ e₃ b₁ b₂ b₃ p₁ p₂ p₃ fe ge Hf Hg. exact (comp_preserves_cartesian Hf Hg). - intros e₁ e₂ b₁ b₂ p₁ p₁' p₂ p₂' fe fe' α β γ H. exact (invertible_2cell_between_preserves_cartesian α β γ 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.