Proposition lwhisker_enriched_profunctor_mor_eq (z : C₃) (y₁ y₂ : C₂) (x : C₁) : lmap_e P y₂ y₁ x #⊗ identity _ · (identity _ #⊗ τ z y₁ · comp_enriched_profunctor_in P Q₂ z x y₁) = sym_mon_braiding _ _ _ #⊗ identity _ · mon_lassociator _ _ _ · identity _ #⊗ rmap_e Q₁ z y₁ y₂ · (identity _ #⊗ τ z y₂ · comp_enriched_profunctor_in P Q₂ z x y₂). Proof. refine (!_). etrans. { rewrite !assoc'. do 2 apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_l. rewrite <- enriched_profunctor_transformation_rmap_e. rewrite tensor_comp_id_l. rewrite !assoc'. rewrite comp_enriched_profunctor_comm'. apply idpath. } rewrite !assoc. apply maponpaths_2. refine (!_). etrans. { rewrite <- tensor_split'. apply tensor_split. } apply maponpaths_2. refine (!_). etrans. { do 4 apply maponpaths_2. rewrite !assoc'. rewrite <- tensor_lassociator. rewrite !assoc. apply maponpaths_2. rewrite tensor_id_id. rewrite <- tensor_split'. apply tensor_split. } refine (_ @ id_right _). rewrite !assoc'. apply maponpaths. etrans. { rewrite !assoc. rewrite <- sym_mon_hexagon_lassociator. rewrite !assoc'. etrans. { do 2 apply maponpaths. rewrite !assoc. rewrite mon_lassociator_rassociator. rewrite id_left. apply idpath. } apply maponpaths. rewrite !assoc. rewrite sym_mon_braiding_inv. apply id_left. } apply mon_lassociator_rassociator. 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.Composition.CompositionProf. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Import MonoidalNotations.