Definition local_fib_weq_local_cleaving {B : bicat} (D : disp_bicat B) : local_cleaving D ≃ local_fib D. Proof. use make_weq. - exact local_cleaving_to_local_fib. - use isweq_iso. + exact local_fib_to_local_cleaving. + abstract (intro HD ; repeat (use funextsec ; intro) ; use total2_paths_f ; [ apply idpath | ] ; cbn ; use total2_paths_f ; [ apply idpath | ] ; cbn ; apply isaprop_is_cartesian_2cell). + abstract (intro HD ; repeat (use funextsec ; intro) ; use total2_paths_f ; [ apply idpath | ] ; cbn ; use total2_paths_f ; [ apply idpath | ] ; cbn ; apply isaprop_is_cartesian). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. 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.TransportLaws. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles.