In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a full and faithful functor. Formal definitions Explicitly, let C and D be (locally small) categories and let F : C → D be a functor from C to D. The functor F induces a function for every pair of objects X and Y in C. The functor F is said to be faithful if FX,Y is injective full if FX,Y is surjective fully faithful (= full and faithful) if FX,Y is bijective for each X and Y in C. A mnemonic for remembering the term "full" is that the image of the function fills the codomain; a mnemonic for remembering the term "faithful" is that you can trust (have faith) that implies . Properties A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D. Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in D not of the form FX for some X in C. Morphisms between such objects clearly cannot come from morphisms in C. A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if F : C → D is a full and faithful functor and then . Examples The forgetful functor U : Grp → Set maps groups to their underlying set, "forgetting" the group operation. U is faithful because two group homomorphisms with the same domains and codomains are equal if they are given by the same functions on the underlying sets. This functor is not full as there are functions between the underlying sets of groups that are not group homomorphisms. A category with a faithful functor to Set is (by definition) a concrete category; in general, that forgetful functor is not full. The inclusion functor Ab → Grp is fully faithful, since Ab (the category of abelian groups) is by definition the full subcategory of Grp induced by the abelian groups. Generalization