Lemma counit_is_z_iso_if_right_adjoint_is_fully_faithful : fully_faithful G -> ∏ x, is_z_isomorphism (ε x). Proof. intros ? ?. apply merely_split_monic_is_epi_to_is_z_iso. - apply counit_is_split_monic_if_right_adjoint_is_full. apply fully_faithful_implies_full_and_faithful; assumption. - apply counit_is_epi_if_right_adjoint_is_faithful. apply fully_faithful_implies_full_and_faithful; assumption. Qed.
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.