Proposition lwhisker_runitor_enriched_profunctor {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} {G : C₃ ⟶ C₄} {EG : functor_enrichment G E₃ E₄} {P : E₁ ↛e E₃} {Q : E₂ ↛e E₄} (τ : enriched_profunctor_square EF EG P Q) : lwhisker_enriched_profunctor_square (runitor_enriched_profunctor P) τ = rwhisker_enriched_profunctor_square (runitor_enriched_profunctor Q) (comp_v_enriched_profunctor_square τ (id_v_enriched_profunctor_square EG)). Proof. use eq_enriched_profunctor_transformation ; intros z x ; cbn. use from_comp_enriched_profunctor_eq. intros y. rewrite !assoc. rewrite runitor_enriched_profunctor_mor_comm. refine (!_). etrans. { apply maponpaths_2. apply comp_v_enriched_profunctor_square_mor_comm. } rewrite !assoc'. etrans. { apply maponpaths. apply (runitor_enriched_profunctor_mor_comm Q). } cbn. rewrite <- (enriched_profunctor_transformation_lmap_e τ). cbn. rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_split. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.whiskering. 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.