Definition local_equivalence_inv_inv2cell {x y : B₁} {f g : F x --> F y} (τ : invertible_2cell f g) : invertible_2cell (local_equivalence_inv_onecell f) (local_equivalence_inv_onecell g). Proof. use make_invertible_2cell. - exact (local_equivalence_inv_twocell τ). - use make_is_invertible_2cell. + exact (local_equivalence_inv_twocell (τ^-1)). + abstract (rewrite <- local_equivalence_inv_twocell_comp ; rewrite vcomp_rinv ; apply local_equivalence_inv_twocell_id). + abstract (rewrite <- local_equivalence_inv_twocell_comp ; rewrite vcomp_linv ; apply local_equivalence_inv_twocell_id). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Properties.