nLab
essentially surjective functor

A functor $F:C\to D$ is essentially surjective if for every object $y$ of $D$, there exists an object $x$ of $C$ and an isomorphism $F(x)\cong y$ in $D$. Sometimes one says “essentially surjective on objects” (because it's a weaking of being surjective on objects), and this is sometimes abbreviated to eso.

In any 2-category there is a notion of eso morphism which generalizes the essentially surjective functors in Cat.