@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.