Definition is_invertible_total_to_disp_weq {x y : C} {xx : D x} {yy : D y} {f g : C⟦x,y⟧} {ff : xx -->[ f ] yy} {gg : xx -->[ g ] yy} (α : f ==> g) (αα : ff ==>[α] gg) : @is_invertible_2cell E (x ,, xx) (y ,, yy) (f,, ff) (g,, gg) (α,,αα) ≃ ∑ (Hα : is_invertible_2cell α), is_disp_invertible_2cell Hα αα. Proof. apply weqimplimpl. 3: apply isaprop_is_invertible_2cell. 3: { apply isofhleveltotal2. { apply isaprop_is_invertible_2cell. } intro Hα. pose (α' := (α,,Hα) : invertible_2cell _ _). apply (isaprop_is_disp_invertible_2cell (x:=α') αα). } { apply is_invertible_total_to_disp. } apply is_invertible_disp_to_total. 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.