Julio Song wrote:

>What Keith's string diagram helped me realize (by completely omitting the bottom-side of the square) is precisely the point that we do not care whether or not there exist independent \$$d \to d'\$$ arrows but merely want to _determine a dependent one_ via manipulating \$$f\$$, \$$h\$$, and \$$g\$$.

That is exactly what the \$$\mathrm{hom}\$$ functor is doing.
Also, my diagram reminds me of a [stalagmite](https://en.wikipedia.org/wiki/Stalagmite).

In fact, you gave me an idea as to how to give a possible formal definition of \$$\mathrm{hom}\$$,

\$\mathrm{hom}(f,g)(h)=\begin{cases} u := g\circ h \circ f & \text{ if } target(f)=source(h) \\\\ & \text{ and } target(h)=source(g)\\\\ & \\\\ \varnothing & \text{ otherwise.} \end{cases} \$