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 $\mathbf C$:

$\map F {g \circ f} = F f \circ F g$

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.

Also see

• Equivalence of Definitions of Contravariant Functor

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