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 $\mathbf{C}$?
 A: For start let explain the notation $\langle -,-\rangle$. In a cartesian category, that is a category with finite products, you can define a family of mappings
$$\langle -,-\rangle \colon \mathbf C[Z,A] \times \mathbf C[Z,B] \to \mathbf C[Z,A \times B]$$
that associate to each pair $a \colon Z \to A$ and $b \colon Z \to B$ the unique morphism $\langle a,b \rangle \colon Z \to A \times B$ that satisfies the property $\pi_A \circ \langle a,b \rangle=a$ and $\pi_B \circ \langle a,b \rangle=b$ (where $\pi_A$ and $\pi_B$ are the projections of the product $A \times B$). 
This basically generalize the canonical tupling operation for sets/types: instead of having a pair elements of the sets/types you gain a pair of generalized elements.
There is more $\langle - , -\rangle$ is actually a natural isomorphism between the two presheaves $\mathbf C[-,A]\times \mathbf C[-,B]$ and $\mathbf C[-,A \times B]$. Since it is a natural isomorphism this allows you to represent every generalized element of $A \times B$, that is an element of $\mathbf C[Z,A \times B]$ for some $Z \in \mathbf C$, as an ordered pair of generalized elements of $A$ and $B$, with the same source.
Once you have this operation you can do most of the stuff you would do in classical set theory (actually in every simple typed lambda calculus), for instance you can define mappings to a product $A \times B$ in a sort of point wise way. This is one of the wonderful consequences of yoneda Lemma. To see how let's look at example you are interested in.
When Awodey says that there is a $t_{A,B}$ such that for each $t_{A,B}\langle a, b \rangle = \langle b, a\rangle$ what actually is happening behind the scene is what follow:


*

*for start you have a switching operation defined on the presheaves $\mathbf C[-,A] \times \mathbf C[-,B]$ and $\mathbf C[-,B]\times \mathbf C[-,A]$, which is an natural isomorphism

*this operation induces a corresponding natural isomorphism between the presheaves $\mathbf C[-,A\times B]$ and $\mathbf C[-,B \times A]$, which are isomorphic to the previous presheaves through the $\langle -,- \rangle$'s

*but by yoneda every natural transformation between presheaves $\mathbf C[-,A \times B]$ and $\mathbf C[-,B \times A]$ is of the form $\mathbf C[-,t]$ for some morphism $t$.


So Awodey's definition of $t_{A,B}$ is basically the following: $t_{A,B}$ is the only morphism that represents the switching operation between the generalized elements of $A \times B$, which can be thought as ordered pairs of generalized elements of $A$ and $B$ by $\langle -, - \rangle$ isomorphisms.
I hope this very long answer can help you seeing some deep insight in the use of generalized elements... when I saw this stuff for the first time I finally understood the power of Yoneda.
A: 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.
A: Yes, they are generalized elements but generalized elements are just arrows (though the term carries some connotations).  For this particular example, you could directly say that it characterizes $\tau$ via $\tau \circ \langle f,g \rangle = \langle g,f \rangle$ for arbitrary arrows $f$ and $g$ of the appropriate types.  In this case we can make a general theorem about cartesian categories that validates the original notation.  In particular, we can say that the internal language (or internal logic or internal type theory) of cartesian categories is one of (multi-sorted) $n$-ary operations.  So we can treat any arrow in the category as a $n$-ary operator and pass parameters around and discard and duplicate parameters as much as we like.  An internal language just means that somewhere someone has shown how to "compile" expressions in the internal language to arrows in the category.
However, in a symmetric monoidal category, the internal language is a (very basic) linear type theory which allows us to define $\tau$ above using the same notation, but $\delta(a) = (a, a)$ would not be allowed.
The links above for internal logic and internal type theory tell you how to what features you can use to define arrows depending on the additional structure your category supports.  For example, a topos validates many of the operations of set theory allowing you to do reasoning very much like you're used to.
A: 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.
