\begin{definition}[Definition:Functor/Contravariant/Definition 1]
Let $\mathbf C$ and $\mathbf D$ be metacategories.
A '''contravariant functor''' $F : \mathbf C \to \mathbf D$ consists of:
* An '''object functor''' $F_0$ that assigns to each object $X$ of $\mathbf C$ an object $FX$ of $\mathbf D$.
* An '''arrow functor''' $F_1$ that assigns to each arrow $f : X \to Y$ of $\mathbf C$ an arrow $Ff : FY \to FX$ of $\mathbf D$.
These '''functors''' must satisfy, for any morphisms $X \stackrel f \longrightarrow Y \stackrel g \longrightarrow Z$ in $\mathbf C$:
:$\map F {g \circ f} = F f \circ F g$
and:
:$\map F {\operatorname {id}_X} = \operatorname {id}_{F X}$
where:
:$\operatorname {id}_W$ denotes the identity arrow on an object $W$
and:
:$\circ$ is the composition of morphisms.
\end{definition}