Proposition rwhisker_enriched_profunctor_mor_eq (z : C₃) (y₁ y₂ : C₂) (x : C₁) : lmap_e P₁ y₂ y₁ x #⊗ identity _ · (τ y₁ x #⊗ identity _ · comp_enriched_profunctor_in P₂ Q z x y₁) = sym_mon_braiding _ _ _ #⊗ identity _ · mon_lassociator _ _ _ · identity _ #⊗ rmap_e Q z y₁ y₂ · (τ y₂ x #⊗ identity _ · comp_enriched_profunctor_in P₂ Q z x y₂). Proof. rewrite !assoc. rewrite <- tensor_comp_id_r. rewrite <- enriched_profunctor_transformation_lmap_e. rewrite tensor_comp_id_r. rewrite !assoc'. rewrite comp_enriched_profunctor_comm. rewrite !assoc. apply maponpaths_2. refine (!_). etrans. { rewrite !assoc'. do 2 apply maponpaths. rewrite <- tensor_split. apply tensor_split'. } rewrite tensor_comp_id_l. rewrite !assoc. apply maponpaths_2. etrans. { rewrite !assoc'. rewrite <- tensor_id_id. rewrite <- tensor_lassociator. rewrite !assoc. rewrite <- tensor_comp_id_r. rewrite <- tensor_sym_mon_braiding. rewrite tensor_comp_id_r. apply idpath. } rewrite !assoc'. apply maponpaths. rewrite sym_mon_tensor_lassociator'. rewrite !assoc'. do 2 apply maponpaths. rewrite mon_rassociator_lassociator. rewrite id_right. 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.Composition.CompositionProf. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Import MonoidalNotations.