Proposition enriched_profunctor_double_bicat_id_comp_square_laws : double_bicat_id_comp_square_laws enriched_profunctor_ver_bicat_sq_id_comp_whisker. Proof. repeat split. - intros C₁ C₂ C₃ C₄ C₅ C₆ C₇ C₈ C₉ P₁ P₂ Q₁ Q₂ R₁ R₂ F₁ F₂ G₁ G₂ H₁ H₂ τ₁ τ₂ θ₁ θ₂. apply enriched_profunctor_interchange. - intro C. use eq_enriched_profunctor_transformation ; intros z x ; cbn. apply idpath. - intros C₁ C₂ C₃ P Q. use eq_enriched_profunctor_transformation ; intros z x ; cbn. use from_comp_enriched_profunctor_eq. intros y. rewrite id_right. rewrite (comp_v_enriched_profunctor_square_mor_comm (id_h_enriched_profunctor_square P) (id_h_enriched_profunctor_square Q)). cbn. rewrite tensor_id_id. apply id_left. - intros C₁ C₂ C₃ P Q. use eq_enriched_profunctor_transformation ; intros z x ; cbn. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. 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. Require Import UniMath.CategoryTheory.EnrichedCats.Profunctors.RepresentableLaws. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.UnivalenceOp. Require Import UniMath.Bicategories.Core.Examples.OpCellBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.EnrichedCats. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.Core. Import MonoidalNotations.