Definition iso_in_E₂_inv {x y : E₂} (f g : x --> y) : invertible_2cell f g → invertible_2cell (mor_E₂_to_E₁ f) (mor_E₂_to_E₁ g). Proof. intros α. use tpair. - exact (cell_E₂_to_E₁ (cell_from_invertible_2cell α)). - use tpair. + exact (cell_E₂_to_E₁ (α^-1)). + split. * exact ((cell_E₂_to_E₁_vcomp α (α^-1)) @ maponpaths cell_E₂_to_E₁ (pr122 α) @ cell_E₂_to_E₁_id₂ _). * exact ((cell_E₂_to_E₁_vcomp (α^-1) α) @ maponpaths cell_E₂_to_E₁ (pr222 α) @ cell_E₂_to_E₁_id₂ _). 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.