Essentially surjective functor

Summary

A functor $F \colon \mathcal{C} \to \mathcal{D}$ is essentially surjective when every object of $\mathcal{D}$ is isomorphic to $F(A)$ for some object $A$ of $\mathcal{C}$. It is surjectivity on objects up to isomorphism, which is the right notion in category theory, where isomorphic objects are interchangeable. A fully faithful and essentially surjective functor is an equivalence of categories.