[Christopher wrote](https://forum.azimuthproject.org/discussion/comment/18902/#Comment_18902):

> Which means given a pointed graph that generates a sequence, we can construct a graph that generates every nth value of the series. Which is pretty non obvious I think!

In some sense, a free category is the union over the sequence of all such graphs! (We don't even have to worry about pointed graphs in particular: powers of the adjacency matrix tell us about the number of paths between every pairs of vertices.)

> Which means given a pointed graph that generates a sequence, we can construct a graph that generates every nth value of the series. Which is pretty non obvious I think!

In some sense, a free category is the union over the sequence of all such graphs! (We don't even have to worry about pointed graphs in particular: powers of the adjacency matrix tell us about the number of paths between every pairs of vertices.)