Definition:Functor/Covariant

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

A covariant 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 $Ff : FX \to FY$ of $\mathbf D$.

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


 * $F(g \circ f) = Fg \circ Ff$, 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.

The behaviour of a covariant functor can be pictured as follows:


 * $\begin{xy}

<4em,4em>*{\mathbf C} = "C", <0em,0em>*+{X} = "a", <4em,0em>*+{Y} = "b", <4em,-4em>*+{Z}= "c",

"a";"b" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f}, "b";"c" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g}, "a";"c" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{g \circ f},

"C"+/r9em/*{\mathbf D}, "C"+/r2em/;"C"+/r6em/ **@{-} ?>*@{>} ?*!/_1em/{F},

"b"+/r2em/+/_2em/;"b"+/r6em/+/_2em/ **@{~} ?>*@2{>} ?<>(.5)*!/_.6em/{F},

"a"+/r13em/*+{FX}="Fa", "b"+/r13em/*+{FY}="Fb", "c"+/r13em/*+{FZ}="Fc",

"Fa";"Fb" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{Ff}, "Fb";"Fc" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{Fg}, "Fa";"Fc" **@{-} ?>*@{>} ?<>(.7)*!/r3em/{F \left({g \circ f}\right) = \\ Fg \circ Ff}, \end{xy}$

Also known as
Many sources simply call this a functor.

Also see

 * Contravariant Functor