Lemma pentagon_identity {a b c d e : C} (k : C ⟦ a, b ⟧) (h : C ⟦ b, c ⟧) (g : C ⟦ c, d ⟧) (f : C ⟦ d, e ⟧) : lassociator k h (g · f) • lassociator (k · h) g f = ((id2 k ⋆ lassociator h g f) • lassociator k (h · g) f) • (lassociator k h g ⋆ id2 f). Proof. rewrite <- lassociator_lassociator. unfold hcomp at 2. rewrite lwhisker_id2. rewrite id2_right. apply maponpaths_2. apply maponpaths_2. unfold hcomp. rewrite id2_rwhisker. apply pathsinv0. apply id2_left. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.Bicategories.WkCatEnrichment.prebicategory. Require Import UniMath.Bicategories.WkCatEnrichment.Notations. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.