Definition LRB_associator_nat_z_iso (x y z w : B) : nat_z_iso (functor_composite (functor_composite (precategory_binproduct_assoc (R (hom (C := B) x y)) (R (hom (C := B) y z)) (R (hom (C := B) z w)) ) (pair_functor (LRB_composition x y z) (functor_identity (R (hom z w))) ) ) (LRB_composition x z w) ) (functor_composite (pair_functor (functor_identity (R (hom (C := B) x y))) (LRB_composition y z w)) (LRB_composition x y w)). Proof. use lift_nat_z_iso_along. - exact ((hom x y) ⊠ ((hom y z) ⊠ (hom z w))). - repeat (apply pair_functor) ; apply (η (hom _ _)). - repeat (apply pair_functor_eso) ; apply eso. - repeat (apply pair_functor_ff) ; apply ff. - exact (LRB_associator_nat_z_iso_pre x y z w). 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.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.PrecompEquivalence. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.RezkCompletions.RezkCompletions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.YonedaLemma. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Properties. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition.