Definition enriched_profunctor_nat_iso_weq_vertible_invertible_2cell {C₁ C₂ : enriched_profunctor_verity_double_bicat} {P Q : C₁ -|-> C₂} (τ : P =|=> Q) : is_iso_enriched_profunctor_transformation τ ≃ is_invertible_2cell τ. Proof. use weqimplimpl. - intros Hτ. use make_is_invertible_2cell. + exact (inv_enriched_profunctor_transformation _ Hτ). + apply inv_enriched_profunctor_transformation_right. + apply inv_enriched_profunctor_transformation_left. - intros Hτ y x. use make_is_z_isomorphism. + exact (pr1 (Hτ^-1) y x). + split. * abstract (exact (from_eq_enriched_profunctor_transformation (vcomp_rinv Hτ) y x)). * abstract (exact (from_eq_enriched_profunctor_transformation (vcomp_linv Hτ) y x)). - apply isaprop_is_iso_enriched_profunctor_transformation. - apply isaprop_is_invertible_2cell. Defined.
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.