Proposition uwhisker_enriched_profunctor_square_id_square {V : benabou_cosmos} {C₁ C₂ : category} {F G : C₁ ⟶ C₂} {τ : F ⟹ G} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} {EF : functor_enrichment F E₁ E₂} {EG : functor_enrichment G E₁ E₂} (Hτ : nat_trans_enrichment τ EF EG) : uwhisker_enriched_profunctor_square Hτ (id_v_enriched_profunctor_square EF) = dwhisker_enriched_profunctor_square Hτ (id_v_enriched_profunctor_square EG). Proof. use eq_enriched_profunctor_transformation ; intros z x ; cbn. rewrite !tensor_linvunitor. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- !tensor_split. rewrite tensor_sym_mon_braiding. rewrite !assoc'. refine (!(id_left _) @ _). rewrite <- mon_lunitor_linvunitor. etrans. { rewrite !assoc'. apply maponpaths. rewrite !assoc. exact (!(Hτ z x)). } rewrite !assoc. do 2 apply maponpaths_2. refine (_ @ id_right _). rewrite <- mon_runitor_rinvunitor. rewrite !assoc. rewrite sym_mon_braiding_runitor. 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.