Full functor

A functor $F:\mathcal{C}\to \mathcal{D}$ is full when each function $\mathcal{C}(A,B)\to \mathcal{D}(FA,FB)$ is surjective. It is a faithful functor when each is injective. A full and faithful functor identifies its source, up to equivalence, with a full subcategory of the target. Fullness only concerns morphisms between image objects; it does not require every target object to lie in the image.