Definition invertible_2cell_from_cartesian (HD : disp_univalent_2_1 D) {x y : B} {f g : x --> y} {α : f ==> g} {Hα : is_invertible_2cell α} {xx : D x} {yy : D y} {ff : xx -->[ f ] yy} (Hff : cartesian_1cell D ff) {gg : xx -->[ g ] yy} {αα : ff ==>[ α ] gg} (Hαα : is_disp_invertible_2cell Hα αα) : cartesian_1cell D gg. Proof. use (invertible_2cell_from_cartesian_help Hff). - exact (isotoid_2_1 (pr2 HB) (make_invertible_2cell Hα)). - refine (disp_isotoid_2_1 _ HD (isotoid_2_1 (pr2 HB) (make_invertible_2cell Hα)) ff gg _). simple refine (transportb (λ z, disp_invertible_2cell z ff gg) (idtoiso_2_1_isotoid_2_1 _ _) (_ ,, _)). + exact αα. + exact Hαα. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Opposite. 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.TransportLaws. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.DispPseudofunctor. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.Display.StrictPseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.StrictPseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Examples.StrictToPseudo. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Projection. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat.