Definition:Functor/Contravariant

Definition 2
Let $\mathbf C$ and $\mathbf D$ be metacategories.

A contravariant functor $F : \mathbf C \to \mathbf D$ is a covariant functor $F: \mathbf C^{\text{op}} \to \mathbf D$, where $\mathbf C^{\text{op}}$ is the dual category of $\mathbf C$.

Equivalence of Definitions
The definitions above are equivalent.

This is shown on Equivalence of Definitions of Contravariant Functor.

Also see

 * Definition:Covariant Functor