Definition disp_invertible_2cell_to_disp_iso_inv {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) : disp_iso_total xx → is_disp_invertible_2cell x xx. Proof. intros z. use tpair. - exact (pr1 z). - split ; cbn. + cbn in *. apply (@transportf_transpose_right _ (λ α : f ==> f, ff ==>[ α] ff)). refine (_ @ fiber_paths (pr1 (pr2 z))). apply (@transportf_paths _ (λ α : f ==> f, ff ==>[ α] ff)). apply C. + apply (@transportf_transpose_right _ (λ α : g ==> g, gg ==>[ α] gg)). refine (_ @ fiber_paths (pr2 (pr2 z))). apply (@transportf_paths _ (λ α : g ==> g, gg ==>[ α] gg)). apply C. 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.