Proposition enriched_profunctor_comp_cylinder_law : comp_cylinder_law enriched_profunctor_ver_bicat_sq_id_comp_whisker. Proof. split. - intros C₁ C₂ C₃ C₄ C₅ C₆ F₁ F₂ F₃ P₁ P₂ P₁' P₂' Q₁ Q₂ Q₁' Q₂' τ₁ τ₂ θ₁ θ₂ s₁ s₁' s₂ s₂'. apply enriched_profunctor_square_comp_cylinder_left. - intros C₁ C₂ C₃ C₄ C₅ C₆ F₁ F₂ F₃ F₄ F₅ F₆ F₇ F₈ P₁ P₂ P₃ τ₁ τ₂ θ₁ θ₂ s₁ s₁' s₂ s₂'. intros p q. pose (H₁ := disp_vcomp2 (disp_lwhisker (pr2 F₆) (pr2 θ₂)) (disp_rwhisker (pr2 F₇) (pr2 θ₁))). pose (H₂ := disp_vcomp2 (disp_lwhisker (pr2 F₂) (pr2 τ₂)) (disp_rwhisker (pr2 F₃) (pr2 τ₁))). refine (_ @ enriched_profunctor_square_comp_cylinder_down _ _ _ _ p q H₁ H₂ @ _). + use eq_enriched_profunctor_transformation. intros x y. apply idpath. + use eq_enriched_profunctor_transformation. intros x y. 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.