Definition local_equivalence_inv_onecell_comp {x y z : B₁} (f : F x --> F y) (g : F y --> F z) : invertible_2cell (local_equivalence_inv_onecell (f · g)) (local_equivalence_inv_onecell f · local_equivalence_inv_onecell g). Proof. refine (comp_of_invertible_2cell (local_equivalence_inv_inv2cell _) (inv_of_invertible_2cell (local_equivalence_unit_inv2cell _))). refine (comp_of_invertible_2cell _ (psfunctor_comp F _ _)). refine (comp_of_invertible_2cell (rwhisker_of_invertible_2cell _ _) (lwhisker_of_invertible_2cell _ _)). - exact (inv_of_invertible_2cell (local_equivalence_counit_inv2cell f)). - exact (inv_of_invertible_2cell (local_equivalence_counit_inv2cell g)). 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.