# 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 $\mathbf C$:

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

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.

### 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 $\mathbf C$.

## Also see

## Co- and Contravariance

Both **covariant** and **contravariant functors** are paramount in all of contemporary mathematics.

The intention behind defining a **functor** is to formalise and abstract the intuitive notion of "preserving structure".

**Functors** thus can be understood as a generalisation of the concept of homomorphism in all its instances.

This explains why one would be led to contemplate **covariant functors**.

However, certain "natural" operations like transposing a matrix do not preserve the structure as rigidly as a homomorphism (we do have Transpose of Matrix Product, however).

Because of the abundant nature of this type of operation, the concept of a **contravariant functors** was invented to capture their behaviour as well.