Full and faithful functors

In category theory, a faithful functor (respectively a full functor) is a functor that is injective (respectively surjective) on hom-sets.

Explicitly, let C and D be (locally small) categories and let F : CD be a functor from C to D. The functor F induces a function

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 : XY 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.