Natural transformations in Awodey's Category Theory Exercise 7.11.8

$Z$ is a given functor, hence it is already defined on objects and arrows; you're baffled by the fact that it is not given explicitly.

But you don't need that: it's good old abstract nonsense :)

Awodey's trick is simple, in hindsight: when you want to prove that two arrows to a product are equal, it's enough to prove that they become equal whenever composed with each of the projection (so, the same argument works even with arbitrarily large products, and with general limits).

What I mean is that the universal property of the product (/the limit) says that $(Ff\times Gf)\circ h_C = h_D\circ Zf$ (i.e. $h$ is natural) iff $p_{1,C}\circ (Ff\times Gf)\circ h_C = p_{1,C}\circ h_D\circ Zf$ and $p_{2,C}\circ (Ff\times Gf)\circ h_C = p_{2,C}\circ h_D\circ Zf$. Basically because there can be a unique arrow with this property, so the two are in fact one.

That's what you are supposed to prove, and that's what Awodey does. Bye!