Essential surjectivity

A functor $F:\mathcal{C}\to \mathcal{D}$ is essentially surjective when every object of $\mathcal{D}$ is isomorphic to $FA$ for some $A\in \mathcal{C}$. Together with full and faithful, this characterizes an equivalence of categories under the appropriate axiom of choice convention. For large categories, choosing a quasi-inverse on all objects requires a universe or a class-choice convention.