Lemma rad_form_adjunction : form_adjunction F rad rad_eta_trans rad_eps_trans. 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.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. 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.Core.Univalence. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Adjunctions.Core.