Lemma are_adjoints_closed_under_iso_data {A B : category} (F G : functor A B) (H : functor B A) (αiso : @z_iso [A,B] F G) (HF : are_adjoints F H) : adjunction_data A B. Proof. set (α := pr1 αiso : nat_trans F G). set (αinv := inv_from_z_iso αiso : nat_trans G F). destruct HF as [[α' β'] [HF1 HF2]]. exists G. exists H. split. - apply (nat_trans_comp _ _ _ α' (post_whisker α H)). - apply (nat_trans_comp _ _ _ (pre_whisker H αinv) β'). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.SplitMonicsAndEpis. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Section HomSetIsoRightFromPartial. End HomSetIsoRightFromPartial. Section HomSetIsoLeftFromPartial. End HomSetIsoLeftFromPartial.