Definition:Functor/Covariant

Definition
Let $\mathcal C$ and $\mathcal D$ be metacategories.

A covariant functor $F : \mathcal C \to \mathcal D$ consists of:


 * An object functor that assigns to each object $X$ of $\mathcal C$ an object $FX$ of $\mathcal D$.


 * An arrow functor that assigns to each arrow $f : X \to Y$ of $\mathcal C$ an arrow $Ff : FX \to FY$ of $\mathcal D$.

These functors must satisfy, for any morphisms $X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z$ in $\mathcal C$:


 * $F(g \circ f) = Fg \circ Ff$, and $F(\operatorname{id}_X) = \operatorname{id}_{FX}$

where $\operatorname{id}_W$ denotes the identity arrow on an object $W$, and $\circ$ is the composition of morphisms.

Also known as
Many sources simply call this a functor.