Julio Song wrote:

>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?

As it stands, that diagrams cannot make \\(? = g\circ h\circ g\\), since \\(f\\) is pointing the wrong way.

\\[
d \overset{f}\leftarrow c \overset{h}\rightarrow c' \overset{g}\rightarrow d'\\\\
\not= \\\\
d \overset{?}\rightarrow d'.
\\]

However, if we use Anindya's diagram,

>\[
\begin{matrix}
& & h & & \\
& c & \rightarrow & c' &\\
f & \uparrow & & \downarrow & g\\
& d & \rightarrow & d' &\\
& & ? & & \\
\end{matrix}
\]

then it's easy to *see* that \\(? = g\circ h\circ g\\) is satisfied, which is the same as saying,
\\[
d \overset{f}\rightarrow c \overset{h}\rightarrow c' \overset{g}\rightarrow d'\\\\
= \\\\
d \overset{?}\rightarrow d'.
\\]

which from Anindya's diagram is very easy to verify.