Every locally small category comes with a very interesting functor indeed. Given objects and we can find the set of morphisms from to . I say that this is a functor .
We’ve given how behaves on pairs of objects. Now if we have morphisms and in we need to construct a function . Notice that the direction of the arrow in the first slot gets reversed, since we want the functor to be contravariant in that place. So, given a morphism we define .
Clearly if we pick a pair of identity morphisms, is the identity function on . Also, if we take , , , and in , then we can check
Again, notice that the order of composition gets swapped in the first slot. Thus is a functor, contravariant in the first slot and covariant in the second.
Not only do hom-functors exist for all locally small categories, but we’ll see that they have all sorts of special properties that will come in handy.