A functor is essentially surjective, or essentially surjective on objects (sometimes abbreviated to eso), if it is surjective on objects “up to isomorphism”.
Any bijective-on-objects functor is essentially surjective.
A composition of any two essentially surjective functors is essentially surjective.
If is essentially surjective, then is essentially surjective.
Strengthening the last example, there is an orthogonal factorization system (in the up-to-isomorphism strict sense) on , in which eso functors are the left class and fully faithful functors are the right class.
This is an “up-to-isomorphism” version of the bo-ff factorization system, which is a 1-categorical orthogonal factorization system on in which the left class consists of bijective-on-objects functors. Thus essentially surjective is a non-evil version of “bijective on objects”, i.e. the version which views as a bicategory.
In particular, while a functor factors uniquely-up-to-isomorphism as a b.o. functor followed by a fully faithful one, it factors only uniquely-up-to-equivalence as an e.s.o. functor followed by a fully faithful one. Since b.o. functors are also e.s.o., any (eso,ff) factorization of some functor is equivalent to its (bo,ff) factorization.
In any 2-category there is a notion of eso morphism which generalizes the essentially surjective functors in Cat. In a regular 2-category, these form a factorization system in a 2-category together with the ff morphisms.