Proposition enriched_lwhisker_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) (Q : E₂ ↛e E₃) {R₁ R₂ : E₃ ↛e E₄} (τ : enriched_profunctor_transformation R₁ R₂) : enriched_profunctor_comp_transformation (lwhisker_enriched_profunctor P (lwhisker_enriched_profunctor Q τ)) (lassociator_enriched_profunctor P Q R₂) = enriched_profunctor_comp_transformation (lassociator_enriched_profunctor P Q R₁) (lwhisker_enriched_profunctor (comp_enriched_profunctor P Q) τ). 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. } refine (!_). etrans. { rewrite !assoc. apply maponpaths_2. apply lassociator_enriched_profunctor_data_comm. } use is_inj_internal_transpose. 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. unfold internal_transpose. rewrite !internal_beta. rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. apply maponpaths. etrans. { rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. apply maponpaths. rewrite !assoc. apply maponpaths_2. rewrite <- tensor_comp_id_r. apply lassociator_enriched_profunctor_mor_comm. } etrans. { unfold lassociator_enriched_profunctor_mor_ob. rewrite !assoc. rewrite sym_mon_braiding_inv. rewrite id_left. rewrite !assoc'. do 2 apply maponpaths. apply (lwhisker_enriched_profunctor_mor_comm (comp_enriched_profunctor P Q)). } refine (!_). etrans. { rewrite !assoc. rewrite <- tensor_comp_id_l. do 2 apply maponpaths_2. apply maponpaths. 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 tensor_comp_id_r. rewrite !assoc'. do 2 apply maponpaths. rewrite !assoc. rewrite <- tensor_comp_id_r. apply lassociator_enriched_profunctor_mor_comm. } unfold lassociator_enriched_profunctor_mor_ob. rewrite !assoc. apply maponpaths_2. refine (!_). etrans. { rewrite !assoc'. rewrite <- tensor_split'. rewrite tensor_split. apply idpath. } rewrite !assoc. apply maponpaths_2. refine (!_). etrans. { rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. rewrite tensor_rassociator. apply idpath. } rewrite !assoc. rewrite sym_mon_braiding_inv. rewrite id_left. rewrite tensor_id_id. 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.