Defining natural transformations based on generalized elements? 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 $\mathbf{C}$?

What Awodey is trying to express in intuitive notation is that the twist map is $\langle \pi_1, \pi_0\rangle:A\times B\to B\times A$, so that $\pi_0\circ t_{(A,B)}=\pi_1$ and $\pi_1\circ t_{(A,B)}=\pi_0$. It's easy to see that for any pair of generalized elements $a:Z\to A$ and $b:Z\to B$ this will give you the property that you cite from Awodey. Conversely, if for all such generalized elements Awodey's property holds, then in particular it holds of $\langle \pi_0,\pi_1\rangle=id_{A\times B}$, and the universal property of products gives us again that $t_{(A,B)}=\langle\pi_1,\pi_0\rangle$.
In any case, what makes sure that $t_{(A,B)}$ exists is the existence of products in $\mathbf{C}$ and their universal property.
