## Hom Functors

Given a category with objects such as: a,b...x,y - f:C(a,b) represents all the morphisms from a to b.
- g:C(x,y) represents all the morphisms from x to y.
We can now take this up one level. A hom functor C(f,g) represents all the functors from f to g. |

We require this hom functor to interact nicely with the hom sets so the diagram on the left needs to commute. |

This gives rise to a bi-functor to the category of sets:

C^{op}×C→Set

A bifunctor is a functor in two arguments, it could be written H(C^{op},C)→Set . That is, it is contravarient in the first argument and covarient in the second).

## Next steps

Hom sets are used in the page about the Yonada Lemma to relate structures at different levels of abstraction.