Proposition runitor_enriched_profunctor_mor_eq (y y' y'' : C₂) (x : C₁) : lmap_e P y'' y' x #⊗ identity _ · (sym_mon_braiding _ _ _ · lmap_e P y' y x) = sym_mon_braiding _ _ _ #⊗ identity _ · mon_lassociator _ _ _ · identity _ #⊗ enriched_comp E₂ y y' y'' · (sym_mon_braiding _ _ _ · lmap_e P y'' y x). Proof. rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. refine (!_). etrans. { do 2 apply maponpaths. rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. rewrite lmap_e_comp'. apply idpath. } rewrite !assoc. do 2 apply maponpaths_2. rewrite sym_mon_tensor_rassociator. do 2 apply maponpaths_2. refine (!(id_right _) @ _). rewrite <- mon_lassociator_rassociator. rewrite !assoc. apply maponpaths_2. etrans. { rewrite !assoc'. apply maponpaths. rewrite !assoc. apply sym_mon_hexagon_lassociator. } rewrite !assoc. rewrite <- tensor_comp_id_r. rewrite sym_mon_braiding_inv. rewrite tensor_id_id. rewrite id_left. apply idpath. 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.StandardProfunctors. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.ProfunctorTransformations. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.ProfunctorSquares. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Invertible. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Composition.CompositionProf. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Import MonoidalNotations.