Every category is the free category for a given graph?

I am wondering if for any category $C$ (at least a small category), we can find a graph $G$ (at least a small graph), such that $C$ is the free category generated by the graph $G$.

I think this result comes in handy, if we want to construct a category, given any other such category $J$, by appending a family of objects and arrows to $J$.

• Every category is the quotient category of a free category. This fits to the usual pattern, first take the free object and then mod out the appropriate relations. – Martin Brandenburg Jun 14 '14 at 13:24

• Of course, I forgot the quotient making $gg^{-1}$ the identity (even in a free group). Sorry for the useless comment. ;) – Pece Jun 15 '14 at 6:04