Definition is_invertible_total_to_disp {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. intros Hαα. refine (is_invertible_total_to_base _ _ Hαα,, is_invertible_total_to_fiber _ _ Hαα). 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.