Anindya Bhattacharyya wrote:

>re **puzzle 109** – I think the practical lesson is that any database is completely specified by what the functor does to the graph. Once we know that, we can extend it to the rest of category by using the two functor laws.

It's not so much that \\(F\\) does something to the graph, but rather that it picks or *instantiates* out a bunch of sets and functions between them that 'look like' the graph in question.
Such sets and their functions are called *instances* in the book (and by Simon Willerton) above.