Definition arrow_subbicat_local_iso_cleaving {B : bicat} (D : arrow_subbicat B) : local_iso_cleaving (disp_map_bicat_to_disp_bicat D). Proof. intros x y f g hx hy hf α. simple refine (_ ,, _). - refine (pr1 hf ,, pr12 hf ,, _) ; cbn. exact (comp_of_invertible_2cell (lwhisker_of_invertible_2cell _ α) (pr22 hf)). - simple refine ((id2 _ ,, _) ,, _) ; cbn. + abstract (rewrite id2_rwhisker, id2_right ; apply idpath). + use is_invertible_to_is_disp_invertible. cbn. is_iso. 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.