Definition disp_invertible_2cell_to_disp_iso_weq {a b : C} {f g : C⟦a,b⟧} {x : invertible_2cell f g} {aa : D a} {bb : D b} {ff : aa -->[ f ] bb} {gg : aa -->[ g ] bb} (xx : ff ==>[ x ] gg) : is_disp_invertible_2cell x xx ≃ disp_iso_total xx. Proof. exists (disp_invertible_2cell_to_disp_iso xx). use isweq_iso. - apply disp_invertible_2cell_to_disp_iso_inv. - intros z. apply subtypePath. + intro; apply isapropdirprod ; apply D. + reflexivity. - intros z. apply subtypePath. + intro; apply isapropdirprod ; apply (total_bicat D). + reflexivity. 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.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Invertible_2cells.