Lemma disp_vassocl {D : disp_prebicat} {a b : C} {f g h k : C⟦a,b⟧} {η : f ==> g} {φ : g ==> h} {ψ : h ==> k} {x : D a} {y : D b} {ff : x -->[f] y} {gg : x -->[g] y} {hh : x -->[h] y} {kk : x -->[k] y} (ηη : ff ==>[η] gg) (φφ : gg ==>[φ] hh) (ψψ : hh ==>[ψ] kk) : (ηη •• φφ) •• ψψ = transportb (λ α, _ ==>[α] _) (vassocl _ _ _) (ηη •• (φφ •• ψψ)). Proof. apply (transportf_transpose_right (P := λ x', _ ==>[x'] _)). apply pathsinv0. etrans. { apply disp_vassocr. } apply maponpaths_2. unfold vassocl. apply pathsinv0, pathsinv0inv0. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence.