Thing about that argument is that it shows that \\(h \mapsto g\circ h\circ f\\) is a _possible_ definition for the hom functor, but it doesn't explain why it's _necessary_. If I understand @Julio correctly that's the nub of his question. It's all very well noting that this definition happens to work neatly, but _why_ this definition and not some other one? I must confess I don't have a simple answer to this, and I suspect the best answer might be something like "this definition is the one that makes the Yoneda Lemma work".
Incidentally I have a similar sense of slight puzzlement over the definition of a natural transformation. I can see how this is a neat way of defining "morphisms between functors", but is it the only way? Is there any way of _deriving_ the definition rather than pulling it out of thin air and checking it works? This might seem kinda pedantic and trivial but I suspect that if we were to try generalising these constructions to higher dimensions, picking the "obvious" answer and checking it might not work.