Proposition enriched_rwhisker_lwhisker {V : benabou_cosmos} {C₁ C₂ C₃ C₄ : category} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} {E₃ : enrichment C₃ V} {E₄ : enrichment C₄ V} (P : E₁ ↛e E₂) {Q₁ Q₂ : E₂ ↛e E₃} (τ : enriched_profunctor_transformation Q₁ Q₂) (R : E₃ ↛e E₄) : enriched_profunctor_comp_transformation (lwhisker_enriched_profunctor P (rwhisker_enriched_profunctor τ R)) (lassociator_enriched_profunctor P Q₂ R) = enriched_profunctor_comp_transformation (lassociator_enriched_profunctor P Q₁ R) (rwhisker_enriched_profunctor (lwhisker_enriched_profunctor P τ) R). Proof. use eq_enriched_profunctor_transformation ; intros z w ; cbn. use from_comp_enriched_profunctor_eq. intros x. etrans. { rewrite !assoc. apply maponpaths_2. apply lwhisker_enriched_profunctor_mor_comm. } etrans. { rewrite !assoc'. apply maponpaths. apply lassociator_enriched_profunctor_data_comm. } etrans. { rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. apply idpath. } refine (!_). etrans. { rewrite !assoc. apply maponpaths_2. apply lassociator_enriched_profunctor_data_comm. } rewrite !assoc'. apply maponpaths. use is_inj_internal_lam. use from_comp_enriched_profunctor_eq. intros y. use internal_funext. intros v h. rewrite !tensor_comp_r_id_r. rewrite !(tensor_split (comp_enriched_profunctor_in _ _ _ _ _) h). rewrite !assoc'. apply maponpaths. clear v h. rewrite !internal_beta. rewrite !assoc. etrans. { apply maponpaths_2. rewrite <- tensor_comp_id_r. apply lassociator_enriched_profunctor_mor_comm. } etrans. { unfold lassociator_enriched_profunctor_mor_ob. rewrite !assoc'. do 3 apply maponpaths. apply (rwhisker_enriched_profunctor_mor_comm (lwhisker_enriched_profunctor P τ)). } etrans. { do 2 apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_r. do 2 apply maponpaths_2. apply lwhisker_enriched_profunctor_mor_comm. } refine (!_). etrans. { rewrite <- tensor_comp_id_r. do 3 apply maponpaths_2. apply rwhisker_enriched_profunctor_mor_comm. } etrans. { rewrite tensor_comp_id_r. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_r. apply lassociator_enriched_profunctor_mor_comm. } unfold lassociator_enriched_profunctor_mor_ob. rewrite tensor_comp_id_r. rewrite !assoc. do 2 apply maponpaths_2. rewrite tensor_sym_mon_braiding. rewrite !assoc'. rewrite tensor_rassociator. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.EnrichedCats.BenabouCosmos. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. 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.Invertible. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.ProfunctorSquares. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Coend. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Composition.CompositionProf. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Composition.Unitors. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Composition.Associators. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.Composition.Whiskering. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Import MonoidalNotations.