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... :-?

\[

\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... :-?