F\colon C(x,y) \to D(F(x), F(y))
More abstractly, we may say a functor is full if it is 1-surjective – or, in simple terms, ‘surjective on morphisms between given objects’. (Note that a functor may be full without being surjective on morphisms, overall, since is allowed to not hit morphisms between objects that are not in the image of .)
Fullness is most important for functors which are also faithful, and full and faithful functors are often called fully faithful. For ordinary functors this may sound odd, because there is no real sense in which “full” modifies “faithful.” However, in some contexts (such as for morphisms in a general 2-category), there is a good notion of “full-and-faithful” or “fully faithful,” but the right notion of “full” alone is not so clear. “Fully faithful” is also sometimes abbreviated to “ff”; see also bo-ff factorization system.
A subcategory is called a full subcategory if its inclusion functor (which is automatically faithful) is also full, and any full and faithful functor exhibits an equivalence of its domain with a full subcategory of its codomain.