Another layman question from me: in this diagram

$\begin{matrix} & & h & & \\\\ & c & \rightarrow & c' &\\\\ f & \downarrow & & \downarrow & g\\\\ & d & \rightarrow & d' &\\\\ & & ? & & \\\\ \end{matrix}$

why must the question mark function be equivalent to some composition of f, h, and g? Can't there simply exist an independent function in \$$\mathcal{C}\$$ from d to d'? I don't understand how the non-composability of f, h, g necessarily blocks this possibility, or say, why the successful mapping from hom(c,c') to hom(d,d') hinges on the commutativity d \$$\rightarrow\$$ d' = \$$g \circ h \circ f\$$ at all. Maybe I'm misunderstanding something fundamental... :-?