Definition right_adjoint_to_is_right_adjoint {C₁ C₂ : bicat_of_univ_cats} {F : C₁ --> C₂} (HF : internal_right_adj F) : is_right_adjoint F. Proof. refine (internal_right_adj_left_adjoint HF ,, _). use make_are_adjoints. - exact (internal_right_adj_unit HF). - exact (internal_right_adj_counit HF). - split. + abstract (intro x ; cbn ; pose (nat_trans_eq_pointwise (pr12 HF) x) as p ; cbn in p ; rewrite !id_left, !id_right in p ; exact p). + abstract (intro x ; cbn ; pose (nat_trans_eq_pointwise (pr22 HF) x) as p ; cbn in p ; rewrite !id_left, !id_right in p ; exact p). 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.