# Essentially surjective functor

In mathematics, specifically in category theory, a functor

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

is essentially surjective (or dense) 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]

## Notes

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

## References

• Mac Lane, Saunders (September 1998). Categories for the Working Mathematician (second ed.). Springer. ISBN 0-387-98403-8.