Essentially surjective functor

In mathematics, specifically in category theory, a functor

${\displaystyle F:C\to D}$

is essentially surjective if each object ${\displaystyle d}$ of ${\displaystyle D}$ is isomorphic to an object of the form ${\displaystyle Fc}$ for some object ${\displaystyle c}$ of ${\displaystyle C}$.

Any functor that is part of an equivalence of categories is essentially surjective. As a partial converse, any full and faithful functor that is essentially surjective is part of an equivalence of categories.¹