Let $F : \mathbf{C} \to \mathbf{D} : G$ be two functors between categories $\mathbf{C}$ and $\mathbf{D}$. A natural transformation $\eta$ from $F$ to $G$ is a collection of morphisms $\eta : FC \to GC$ in $\mathbf{D}$ for each $C \in \mathbf{C}$.

Particularly, since $FC$ and $GC$ are just objects, we can't simply suppose that they, too, have objects. And sometimes this makes it very difficult for me to come up with a definition if I am working with an arbitrary category $\mathbf{C}$ which I know nothing about.

However, in Awodey's *Category Theory* (p.158), where $\mathbf{C}$ is an arbitrary category with producs, he defines the component of a "twist" natural transformation
$$ t\_{(A,B)} : A \times B \to B \times A$$
by
$$ t\_{(A,B)} \langle a,b \rangle = \langle b,a \rangle$$
without making it explicit what $\langle a,b \rangle$ really are here.

Are they generalized objects $a : Z \to A$ and $b: Z \to B$?

If this is correct, I wonder when is it acceptable to define a component based on its action on generalized elements?
Moreover, how do I know that this morphism really exist in $\mathbf{C}$?