# Definition:Functor/Contravariant

## Definition

### 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 $\mathbf C$:

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

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.

### 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 $\mathbf C$.

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.