**Exercise 4.1.31**

**Theorem**:
Let be the functor sending a small category to the set of all its maps.
is representable.

*Proof*:
Denote by the category with two objects and one non-identity arrow between them.
We will denote the objects by and and the map by .
We claim .
Indeed, we define a natural transformation .
For a category , define as follows.
For a map in , let be the obvious functor sending to , to , and the to .
To check naturality, suppose we have a functor .
We must have .
Note that is essentially (the restriction of to the maps of the category), and is left-composition by .
Start with a map in .
Itâ€™s straightforward to see that applying the left and right side result in the same functor .
Applying the right side to yields:

while applying the left side to yields:

so is natural and .