Proposition comp_v_enriched_profunctor_square_uwhisker {V : benabou_cosmos} {C₁ C₂ C₃ C₄ C₅ C₆ : category} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} {E₃ : enrichment C₃ V} {E₄ : enrichment C₄ V} {E₅ : enrichment C₅ V} {E₆ : enrichment C₆ 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} {τ : G₂ ⟹ G₁} (Hτ : nat_trans_enrichment τ EG₂ EG₁) {H : C₅ ⟶ C₆} {EH : functor_enrichment H E₅ E₆} {P₁ : E₁ ↛e E₃} {P₂ : E₃ ↛e E₅} {Q₁ : E₂ ↛e E₄} {Q₂ : E₄ ↛e E₆} (θ₁ : enriched_profunctor_square EF EG₁ P₁ Q₁) (θ₂ : enriched_profunctor_square EG₂ EH P₂ Q₂) : comp_v_enriched_profunctor_square θ₁ (uwhisker_enriched_profunctor_square Hτ θ₂) = comp_v_enriched_profunctor_square (dwhisker_enriched_profunctor_square Hτ θ₁) θ₂. Proof. use eq_enriched_profunctor_transformation ; intros z v ; cbn. use from_comp_enriched_profunctor_eq. intro x. etrans. { apply comp_v_enriched_profunctor_square_mor_comm. } refine (!_). etrans. { apply comp_v_enriched_profunctor_square_mor_comm. } cbn. rewrite !tensor_comp_r_id_r. rewrite !tensor_comp_l_id_r. rewrite !assoc'. apply maponpaths. rewrite comp_enriched_profunctor_comm. rewrite !assoc. apply maponpaths_2. rewrite tensor_comp_id_l. rewrite !assoc. apply maponpaths_2. etrans. { rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_lassociator. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_sym_mon_braiding. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite tensor_lassociator. rewrite !assoc'. rewrite <- tensor_comp_id_l. rewrite tensor_sym_mon_braiding. rewrite tensor_comp_id_l. apply idpath. } rewrite !assoc. apply maponpaths_2. etrans. { rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite sym_mon_hexagon_lassociator. rewrite !assoc'. rewrite <- tensor_comp_id_l. rewrite sym_mon_braiding_inv. rewrite tensor_id_id. rewrite id_right. apply idpath. } rewrite !assoc. rewrite <- tensor_comp_id_r. rewrite sym_mon_braiding_linvunitor. rewrite mon_inv_triangle. 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.