Definition sigma_disp_invertible_to_pair_disp_invertible {x y : C} {f : C ⟦ x, y ⟧} {xx : (sigma_bicat C D₁ D₂) x} {yy : (sigma_bicat C D₁ D₂) y} {ff1 : pr1 xx -->[ f] pr1 yy} (ff2 : pr2 xx -->[ f,, ff1] pr2 yy) {gg1 : pr1 xx -->[ f] pr1 yy} (gg2 : pr2 xx -->[ f,, gg1] pr2 yy) : @disp_invertible_2cell C (sigma_prebicat C D₁ D₂) _ _ _ _ _ _ (id2_invertible_2cell f) (ff1,, ff2) (gg1,, gg2) → (∑ (p : disp_invertible_2cell (id2_invertible_2cell f) ff1 gg1), disp_invertible_2cell (iso_in_E_weq _ _ (id2_invertible_2cell f ,, p)) ff2 gg2). Proof. intros p. use tpair. - use tpair. + exact (pr11 p). + simple refine (_ ,, (_ ,, _)). * exact (pr1 (disp_inv_cell p)). * apply HD₁. * apply HD₁. - use tpair. + exact (pr21 p). + simple refine (_ ,, (_ ,, _)). * exact (pr2 (disp_inv_cell p)). * apply HD₂. * apply HD₂. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.