\begin{definition}[Definition:Composition Functor on Slice Categories]
Let $\mathbf C$ be a metacategory.
Let $C$ and $D$ be objects of $\mathbf C$.
Let $\mathbf C / C$ and $\mathbf C / D$ be the associated slice categories.
Let $g: C \to D$ be a morphism of $\mathbf C$.
Then $g$ defines a '''composition functor''' $g_* : \mathbf C / C \to \mathbf C / D$:
{{begin-axiom}}
{{axiom|lc= Object functor:
       |m = g_* f := g \circ f
       |rc= The composition $\circ$ is taken in $\mathbf C$
}}
{{axiom|lc= Morphism functor:
       |m = g_* a := a
}}
{{end-axiom}}
That it is in fact a functor is shown on Composition Functor on Slice Categories is Functor.
The effect of $g_*$ is captured in the following commutative diagram:
::$\begin{xy}
<-3em,0em>*+{X} = "X",
<3em,0em>*+{X'} = "X2",
<0em,-4em>*+{C} = "C",
<0em,-8em>*+{D} = "D",
"X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a},
"X";"C" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{f},
"X2";"C" **@{-} ?>*@{>} ?<>(.4)*!/_.6em/{f'},
"C";"D" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{g},
"X";"D" **\crv{<-5em,-4em>} ?>*@{>} ?*!/^1.6em/{g_* f = \\ g \circ f},
"X2";"D" **\crv{<5em,-4em>} ?>*@{>} ?*!/_1.6em/{g_* f' = \\ g \circ f'},
\end{xy}$
\end{definition}