@John wrote:

> this 'complete freedom' in choosing sets for nodes and functions for edges is why \$$\mathbf{Free}(G)\$$ is called the 'free category' on the graph \$$G\$$

One slightly pedantic caveat – we don't quite have "complete freedom" in choosing sets for nodes – if \$$a\$$ is an arrow from node \$$s\$$ to node \$$t\$$, and we choose \$$F(t)\$$ to be the empty set, then we are forced to have \$$F(s)\$$ empty too, with \$$F(a)\$$ the empty function.