Proposition profunctor_square_to_enriched_nat_trans_left_point {V : benabou_cosmos} {C₁ C₂ : category} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} {F G : C₁ ⟶ C₂} {EF : functor_enrichment F E₁ E₂} {EG : functor_enrichment G E₁ E₂} {τ : F ⟹ G} (Hτ : nat_trans_enrichment τ EF EG) (x : C₁) : square_to_enriched_nat_trans_data (uwhisker_enriched_profunctor_square Hτ (id_v_enriched_profunctor_square _)) x = τ x. Proof. cbn. refine (_ @ enriched_to_from_arr E₂ _). apply maponpaths. rewrite !assoc. rewrite functor_enrichment_id. rewrite tensor_linvunitor. rewrite !assoc'. etrans. { apply maponpaths. rewrite !assoc. rewrite <- tensor_split. rewrite tensor_split'. rewrite !assoc'. rewrite <- enrichment_id_right. apply idpath. } rewrite tensor_runitor. rewrite !assoc. rewrite mon_linvunitor_I_mon_rinvunitor_I. rewrite mon_rinvunitor_runitor. apply id_left. 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.