proposition: Lemma rad_form_adjunction : form_adjunction F rad rad_eta_trans rad_eps_trans.
proof: Proof. split; simpl. - intro a. cbn. unfold rad_eta. inv_functor a (rad_ob (F a)). apply z_iso_after_z_iso_inv. - intro b. apply (invmaponpathsweq (weq_from_fully_faithful HF (rad_ob b) (rad_ob b))). simpl; rewrite functor_comp. unfold rad_eta. inv_functor (rad_ob b) (rad_ob (F (rad_ob b))). unfold rad_mor. inv_functor (rad_ob (F (rad_ob b))) (rad_ob b). repeat rewrite assoc. rewrite z_iso_after_z_iso_inv. rewrite <- assoc. rewrite z_iso_inv_after_z_iso. rewrite id_left. rewrite functor_id. apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Adjunctions.Core