Proposition comp_enriched_profunctor_comm (y₁ y₂ : C₂) : lmap_e P y₂ y₁ x #⊗ identity _ · comp_enriched_profunctor_in y₁ = mon_lassociator _ _ _ · sym_mon_braiding V _ _ · mon_lassociator _ _ _ · identity _ #⊗ (sym_mon_braiding V _ _ · rmap_e Q z y₁ y₂) · comp_enriched_profunctor_in y₂. Proof. refine (!(id_left _) @ _). rewrite <- mon_lassociator_rassociator. rewrite !assoc'. apply maponpaths. rewrite !assoc. exact (enriched_profunctor_coend_comm comp_enriched_profunctor_coend y₁ y₂). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Coends. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.EnrichedCats.BenabouCosmos. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Profunctor. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Coend. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Import MonoidalNotations.