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In mathematics, specifically in category theory, a functor

is **essentially surjective** if each object of is isomorphic to an object of the form for some object of .

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]}

**^**Mac Lane (1998), Theorem IV.4.1

- Mac Lane, Saunders (September 1998).
*Categories for the Working Mathematician*(second ed.). Springer. ISBN 0-387-98403-8. - Riehl, Emily (2016).
*Category Theory in Context*. Dover Publications, Inc Mineola, New York. ISBN 9780486809038.

- Essentially surjective functor at the
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