Definition is_disp_invertible_to_is_invertible {x y : B} {f g : x --> y} {α : f ==> g} (Hα : is_invertible_2cell α) {hx : disp_map_bicat_to_disp_bicat x} {hy : disp_map_bicat_to_disp_bicat y} {hf : hx -->[ f ] hy} {hg : hx -->[ g ] hy} (αα : hf ==>[ α ] hg) (Hαα : is_disp_invertible_2cell Hα αα) : is_invertible_2cell (pr1 αα). Proof. use make_is_invertible_2cell. - exact (pr11 Hαα). - abstract (exact (maponpaths pr1 (pr12 Hαα) @ transportb_disp_map_bicat_cell _ _)). - abstract (exact (maponpaths pr1 (pr22 Hαα) @ transportb_disp_map_bicat_cell _ _)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.StreetFibration. 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.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Logic.DisplayMapBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Codomain. Require Import UniMath.Bicategories.Limits.Pullbacks.