# 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* : *C* → *D* 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* : *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*.