A (covariant) functor between categories assigns objects to objects and morphisms
to morphisms, preserving identity morphisms
and composition of morphisms. A covariant
functor 
sends 
to 
and satisfies 
for every composable pair of
morphisms, and 
, where 
is the identity morphism of
(Stacks Project Authors 2026). A contravariant functor reverses the directions
of morphisms and their order in composition
of morphisms, so 
.
Functor
See also
Additive Functor, Category, Contravariant Functor, Covariant Functor, Eilenberg-Steenrod Axioms, Forgetful Functor, Monoidal Functor, Morphism, Schur Functor, Tensor Product FunctorExplore with Wolfram|Alpha
References
The Stacks Project Authors. "Definitions." Definition 4.2.8 in The Stacks Project, Tag 001B, 2026. https://stacks.math.columbia.edu/tag/001B.Referenced on Wolfram|Alpha
FunctorCite this as:
Weisstein, Eric W. "Functor." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Functor.html