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 $\mathbf{C}$?

Yes, $a:Z \to A,\, b : Z \to B$ are generalized elements. Then, $(a,b) : Z \to A \times B$ denotes the unique generalized element with $p_A \circ (a,b) = a$ and $p_B \circ (a,b)=b$ - which exists by the universal property of a product. The twist morphism is defined by $t_{A,B} \circ (a,b) = (b,a)$. This is a rigorous equation.
