## Hom functors

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.

[…] Now that we have a handle on hom functors, we can use them to define other […]

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[…] when we have extra commuting actions on . The complication is connected to the fact that the hom functor is contravariant in its first argument, which if you don’t know much about category theory […]

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