Proposition uwhisker_enriched_profunctor_square_comp {V : benabou_cosmos} {C₁ C₂ C₃ C₄ : category} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} {E₃ : enrichment C₃ V} {E₄ : enrichment C₄ V} {F₁ F₂ F₃ : C₁ ⟶ C₂} {EF₁ : functor_enrichment F₁ E₁ E₂} {EF₂ : functor_enrichment F₂ E₁ E₂} {EF₃ : functor_enrichment F₃ E₁ E₂} {G : C₃ ⟶ C₄} {EG : functor_enrichment G E₃ E₄} {P : E₁ ↛e E₃} {Q : E₂ ↛e E₄} {τ₁ : F₁ ⟹ F₂} (Hτ₁ : nat_trans_enrichment τ₁ EF₁ EF₂) {τ₂ : F₂ ⟹ F₃} (Hτ₂ : nat_trans_enrichment τ₂ EF₂ EF₃) (θ : enriched_profunctor_square EF₁ EG P Q) : uwhisker_enriched_profunctor_square (comp_trans_enrichment Hτ₁ Hτ₂) θ = uwhisker_enriched_profunctor_square Hτ₂ (uwhisker_enriched_profunctor_square Hτ₁ θ). Proof. use eq_enriched_profunctor_transformation ; intros z x ; cbn. rewrite !assoc'. do 2 apply maponpaths. rewrite enriched_from_arr_comp. etrans. { do 3 apply maponpaths_2. rewrite tensor_split. rewrite !assoc. rewrite <- tensor_linvunitor. apply idpath. } rewrite !tensor_comp_id_r. rewrite !assoc'. apply maponpaths. rewrite rmap_e_comp. rewrite !assoc. apply maponpaths_2. rewrite tensor_linvunitor. rewrite <- mon_linvunitor_triangle. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_lassociator. rewrite !assoc'. apply maponpaths. rewrite tensor_id_id. 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.