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 $J$.
No, obviously not. Take, for instance, any non-trivial group considered as a one-object category. More generally, one observes that the only isomorphisms in a category freely generated by a graph are the identities.