Theorem: Let be the functor sending a small category to the set of all its maps.
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:</script>
f \enspace \mapsto \enspace \left{ \begin{array}{l} \circ \mapsto A \
- \mapsto A' \ m \mapsto f \end{array} \right. \enspace \mapsto \enspace \left{ \begin{array}{l} \circ \mapsto F(A) \
- \mapsto F(A') \
m \mapsto F(f)
\end{array} \right.
$$</li> </ul>
while applying the left side to yields:
so is natural and .