Lemma prewhisker_LRB_associator_co {x y z w : B} (f : B ⟦ x, y ⟧) (g : B ⟦ y, z ⟧) (h : B ⟦ z, w ⟧) : # (LRB_composition x z w) (pr1 (LRB_composition_comm x y z) (f : hom _ _, g : hom _ _) #, id₁ (η (hom z w) h) ) · pr1 (LRB_composition_comm x z w) (f · g : hom _ _, h : hom _ _) · # (η (hom x w)) (rassociator f g h) = ((LRB_associator_nat_z_iso x y z w) (η_{x,y} f, (η_{y,z} g, η (hom z w) h))) · # (LRB_composition x y w) (make_dirprod (id₁ (η_{x,y} f)) (pr1 (LRB_composition_comm y z w) (g : hom _ _, h : hom _ _)) : R (hom x y) ⊠ R (hom y w)⟦(_,_),(_,_)⟧ ) · pr1 (LRB_composition_comm x y w) (f : hom _ _, g · h: hom _ _). Proof. set (l := prewhisker_LRB_associator f g h). transparent assert (l1 : (is_z_isomorphism (#η_{ x, w} (rassociator f g h)))). { use functor_on_is_z_isomorphism. apply is_z_iso_rassociator. } apply pathsinv0. use (z_iso_inv_on_left _ _ _ _ (z_iso_inv (_,,l1))). transparent assert (l2 : (is_z_isomorphism (LRB_associator_nat_z_iso x y z w (η_{ x, y} f, (η_{ y, z} g, η_{ z, w} h))))). { apply LRB_associator_nat_z_iso. } rewrite ! assoc'. apply pathsinv0, (z_iso_inv_on_right _ _ _ (z_iso_inv (_,,l2))). rewrite ! assoc' in l. exact l. 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.