\begin{definition}[Definition:Functor/Covariant]
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 $\mathbf C$:
:$\map F {g \circ f} = F g \circ F f$
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.
The behaviour of a '''covariant functor''' can be pictured as follows:
::$\begin{xy}
<4em,4em>*{\mathbf C} = "C",
<0em,0em>*+{X} = "a",
<4em,0em>*+{Y} = "b",
<4em,-4em>*+{Z}= "c",
"a";"b" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f},
"b";"c" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g},
"a";"c" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{g \circ f},
"C"+/r9em/*{\mathbf D},
"C"+/r2em/;"C"+/r6em/ **@{-} ?>*@{>} ?*!/_1em/{F},
"b"+/r2em/+/_2em/;"b"+/r6em/+/_2em/ **@{~} ?>*@2{>} ?<>(.5)*!/_.6em/{F},
"a"+/r13em/*+{FX}="Fa", 
"b"+/r13em/*+{FY}="Fb", 
"c"+/r13em/*+{FZ}="Fc",
"Fa";"Fb" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{Ff},
"Fb";"Fc" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{Fg},
"Fa";"Fc" **@{-} ?>*@{>} ?<>(.7)*!/r3em/{F \left({g \circ f}\right) = \\ Fg \circ Ff},
\end{xy}$
\end{definition}