Lemma LRB_associator_pre_simpl {x y z w : B} (f : B⟦x,y⟧) (g : B⟦y,z⟧) (h : B⟦z,w⟧) : pr1 (LRB_associator_nat_z_iso_pre x y z w) (f : hom _ _, (g : hom _ _, h : hom _ _)) = LRB_associator_pre_simpl_mor f g h. Proof. cbn. unfold LRB_associator_pre_simpl_mor. rewrite ! id_left. rewrite ! id_right. rewrite ! assoc. rewrite id2_left. do 4 (apply maponpaths_2). apply maponpaths. cbn. use total2_paths_f. - cbn ; refine (_ @ id_left _). apply maponpaths_2. etrans. { apply maponpaths, binprod_id. } apply functor_id. - now rewrite transportf_const. Qed.
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.