**Exercise 4.1.28**

For prime, let be a functor sending a a group to its set of elements having order or . For a group homomorphism , simply let be the restriction of to – this is well defined since is prime, and thus necessarily sends an element of order to an element of order or .

**Theorem**:
is representable.
In particular, is isomorophic to .

*Proof*:
Define a natural transformation as follows.
For a group , let send an element of order or to the unique homomorphism such that .
In other words, is the map .
We now check naturality.
Suppose is a homomorphism.
We must have .
Recall that is just left-composition by .
Take a in .
On the right side, we have , then left-composing with yields .
On the left, we have .
Then, .
So both sides yield the same map , verifying that is natural.
So .