Contravariant functor

Covariance and contravariance

Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a covariant functor on the opposite category${\displaystyle C^{\mathrm {op} }}$. Some authors prefer to write all expressions covariantly. That is, instead of saying ${\displaystyle F:C\to D}$ is a contravariant functor, they simply write ${\displaystyle F:C^{\mathrm {op} }\to D}$ (or sometimes ${\displaystyle F:C\to D^{\mathrm {op} }}$) and call it a functor.