\begin{definition}[Definition:Free Category]
Let $G$ be a digraph.
The '''free category''' on $G$, denoted $\map {\mathbf C} G$, is the category with:
{{DefineCategory
| ob = The vertices of $G$
| mor = The walks in $G$
| comp = concatenation of walks
| id = $\operatorname{id}_v$ is the empty walk at $v$
}}
\end{definition}