Essentially surjective functor

In mathematics, specifically in category theory, a functor

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

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

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