\begin{definition}[Definition:Coslice Category]
Let $\mathbf C$ be a metacategory.
Let $C \in \mathbf C_0$ be an object of $\mathbf C$.
The '''coslice category''' of $\mathbf C$ under $C$, denoted $C / \mathbf C$, is the category with:
{{DefineCategory
|ob = $f: C \to X$, i.e. the morphisms of $\mathbf C$ with domain $C$
|mor  = $a: f \to f'$, for all morphisms $a \in \mathbf C_1$ with $a \circ f' = f$
|comp = $a \circ b$ is defined precisely as in $\mathbf C$
|id = $\operatorname{id}_f := \operatorname{id}_X$, for $f: C \to X$
}}
The morphisms can be displayed using a commutative diagram as follows:
::$\begin{xy}
<-3em,0em>*+{X} = "X",
<3em,0em>*+{X'} = "X2",
<0em,4em>*+{C} = "C",
"X";"X2" **@{-} ?>*@{>} ?*!/^1em/{a},
"C";"X" **@{-} ?>*@{>} ?*!/^.6em/{f},
"C";"X2" **@{-} ?>*@{>} ?<>(.6)*!/_1em/{f'},
\end{xy}$
\end{definition}