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.