Definition profunctor_nat_iso_weq_vertible_invertible_2cell {C₁ C₂ : univcat_profunctor_verity_double_bicat} {P Q : C₁ -|-> C₂} (τ : P =|=> Q) : is_profunctor_nat_iso τ ≃ is_invertible_2cell τ. Proof. use weqimplimpl. - intros Hτ. use make_is_invertible_2cell. + exact (inv_profunctor_nat_trans Hτ). + apply inv_profunctor_nat_trans_right. + apply inv_profunctor_nat_trans_left. - intros Hτ y x. use isweq_iso. + intros h. exact ((Hτ^-1 : profunctor_nat_trans _ _) y x h). + abstract (intros h ; cbn ; exact (profunctor_nat_trans_eq_pointwise (vcomp_rinv Hτ) h)). + abstract (intros h ; cbn ; exact (profunctor_nat_trans_eq_pointwise (vcomp_linv Hτ) h)). - apply isaprop_is_profunctor_nat_iso. - apply isaprop_is_invertible_2cell. Defined.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.Categories.HSET.SetCoends. Require Import UniMath.CategoryTheory.Profunctors.Core. Require Import UniMath.CategoryTheory.Profunctors.Examples. Require Import UniMath.CategoryTheory.Profunctors.Squares. Require Import UniMath.CategoryTheory.Profunctors.Transformation. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.ProfunctorTwosidedDispCat. 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.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DoubleCategories.DoubleBicat.Core.