Definition right_adjoint_weq_is_right_adjoint {C₁ C₂ : bicat_of_univ_cats} (F : C₁ --> C₂) : internal_right_adj F ≃ is_right_adjoint F. Proof. use make_weq. - exact right_adjoint_to_is_right_adjoint. - use isweq_iso. + exact is_right_adjoint_to_right_adjoint. + abstract (intro HF ; use subtypePath ; [ intro ; apply isapropdirprod ; apply cellset_property | ] ; apply idpath). + abstract (intro HF ; refine (maponpaths (λ z, _ ,, z) _) ; use subtypePath ; [ intro ; apply isaprop_form_adjunction | ] ; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.UnivalenceOp. Require Import UniMath.Bicategories.Core.Examples.OpMorBicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.Morphisms.Examples.MorphismsInOp1Bicat.