Lemma fully_faithful_conservative {C D : category} (F : functor C D) (FF : fully_faithful F) : conservative F. Proof. unfold conservative. intros a b f is_iso_Ff. use transportf. - exact (FF^-1 (# F f)). - apply fully_faithful_inv_hom_is_inv. - apply (fully_faithful_reflects_iso_proof _ _ _ _ _ _ (_,,is_iso_Ff)). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.Core.Univalence.