Lemma faithful_reflects_mono {C D : category} (F : functor C D) (FF : faithful F) : reflects_morphism F (@isMonic). Proof. unfold reflects_morphism. intros ? ? ? is_monic_Ff. intros ? ? ? eqcomp. apply (Injectivity (# F)). - apply isweqonpathsincl, FF. - apply is_monic_Ff. refine (!(functor_comp F g f) @ _). refine (_ @ functor_comp F h f). apply maponpaths; assumption. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Categories.HSET.Core. 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.covyoneda. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Subcategory.Core.