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)