I have simply been cheating, Fredrick Eisele. The fact that natural transformations and composition give a category implies that any square you can come up with has to commute. Or, put another way, eithet at least one of the transformations you gave isn't in fact a natural transformation, or they're equal.
Argh! Of course I didn't mean to talk about commutativity! It's all about commuting diagrams! And if you look at Fredrick's latest comment you can see that he writes:
> α = c o a, β = c o b
proposing that they're not equal.
I'm sorry for the slightly excessive delay on, my part, I couldn't quite figure out what was going awry, and then forgot about it.