Naturality of bijection given by Yoneda Lemma I'm reading through Wikipedia's proof of the Yoneda Lemma (https://en.wikipedia.org/wiki/Yoneda_lemma), and am having trouble understanding what naturality means in this context.
The articles proves that $Nat(h^A, F) \cong F(A)$. It then goes on to state that if both both sides are seen as functors from $Set^C \times C$ to $Set$, then the bijection provided by the lemma is natural. Other proofs say something like "The bijection is natural in both $F$ and $A$".
My problem is that I don't see what a morphism from $Set^C \times C$ should be mapped to by either of the functors. The functor on the right hand side would need to take a morphism from $f:A\rightarrow B$ from $C$ and a natural transformation $\sigma:F \rightarrow B$  and map them to a function from $F(A)$ to $G(B)$. I don't really see a canonical way of doing so, can somebody help me?

Since others have given detailed answers, at appropriate math.se length, I won't try to do that. But as Clive says, the details can seem reasonably grim at first sight. So I'll just add that if you want a more slow-motion version, I have a bash at making the full-dress Yoneda Lemma, including the naturality claims, as clear as I can in these draft Notes on Basic Category Theory, esp Ch. 11.  
