The second chapter of Awodey’s book is called ‘Abstract Structures’. It gives the usual abstract category-theoretic definitions of epis and monos, of sections and retractions, of initial and terminal objects, of products, and so on. This would certainly be tough going if it was the first time you’d ever encountered these notions. Even as revision/consolidation it’s a bit of a bumpy ride. But for all that, I did get a fair bit out the chapter (Awodey’s clusters of illustrative examples can be very illuminating).
One query. In the sections on products, Awodey starts carefully, talking of a product as an object together with a pair of arrows, and rightly referring to the object A x B as part of a product. And mostly what he says about products reflects this understanding of what products are. But on p. 42 he says that any object A is the unary product of A with itself one time. Is that right? The unary product is surely not just the object but the object with its self-identity arrow.
And one suggestion. The first stage of the two-stage proof at the top of p.27 is surely unnecessarily. Just start in f(-n) = f(-n) * u = f(-n) * g(0) = f(-n) * g(n + –n) etc. [Actually the first stage too illustrates one of Awodey’s quirks, a tendency to occasionally slightly abuse notation without explanation.]
I don’t know the book but some category theorists identify objects with the arrows that map them to themselves so that might explain it.