For those of you working out the homfunctor and why the first coordinate needs to be contravariant, I drew out the simple example used above in my comment #23. It might help calculate how this functor is working. The left diagram shows the how it maps the objects and the right diagram shows how it maps the morphisms. Let me know if there are mistakes.

![How Homfunctor maps objects and morphisms](http://aether.co.kr/images/homfunctor_ob_mor.svg)

Edit: Now that I think about it, this is so natural since morphisms and compositions have directionality which puts the start and target on opposite ends of the morphism and composition must start where the previous left off.

![How Homfunctor maps objects and morphisms](http://aether.co.kr/images/homfunctor_ob_mor.svg)

Edit: Now that I think about it, this is so natural since morphisms and compositions have directionality which puts the start and target on opposite ends of the morphism and composition must start where the previous left off.