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 . It’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 .