Definition:Functor/Contravariant/Definition 1

Definition
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 $\mathbf C$:


 * $F \left({g \circ f}\right) = Ff \circ Fg$

and:
 * $F \left({\operatorname{id}_X}\right) = \operatorname{id}_{FX}$

where:
 * $\operatorname{id}_W$ denotes the identity arrow on an object $W$

and:
 * $\circ$ is the composition of morphisms.

Also see

 * Equivalence of Definitions of Contravariant Functor