Keith wrote:
>\\[
>\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}
>\\]

Thanks for this. Gave me a better perspective on how the hom functor works.

Below is a diagram showing preservation of composition highlighting your hom gadget.

![homfunctor preservation of composition](http://aether.co.kr/images/homfunctor_composition.svg)