Proposition rwhisker_dwhisker_enriched_profunctor_square {V : benabou_cosmos} {C₁ C₂ C₃ C₄ : category} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} {E₃ : enrichment C₃ V} {E₄ : enrichment C₄ V} {F : C₁ ⟶ C₂} {EF : functor_enrichment F E₁ E₂} {G₁ G₂ : C₃ ⟶ C₄} {EG₁ : functor_enrichment G₁ E₃ E₄} {EG₂ : functor_enrichment G₂ E₃ E₄} {τ₁ : G₁ ⟹ G₂} (Hτ₁ : nat_trans_enrichment τ₁ EG₁ EG₂) {P : E₁ ↛e E₃} {Q₁ Q₂ : E₂ ↛e E₄} (τ₂ : enriched_profunctor_transformation Q₁ Q₂) (θ : enriched_profunctor_square EF EG₂ P Q₁) : rwhisker_enriched_profunctor_square τ₂ (dwhisker_enriched_profunctor_square Hτ₁ θ) = dwhisker_enriched_profunctor_square Hτ₁ (rwhisker_enriched_profunctor_square τ₂ θ). Proof. use eq_enriched_profunctor_transformation ; intros z x ; cbn. rewrite !assoc'. apply maponpaths. rewrite <- enriched_profunctor_transformation_lmap_e. rewrite !assoc. apply maponpaths_2. rewrite tensor_linvunitor. rewrite !assoc'. rewrite <- tensor_split. rewrite <- tensor_split'. 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.SquareComp. 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.