> **Puzzle 102.** What is the category with the fewest morphisms that is not a preorder?

Inspired by [Jonathan's answer to puzzle 97](https://forum.azimuthproject.org/discussion/comment/18696/#Comment_18696),
I would say the category with a single object, \\(\star\\), and two morphisms, \\(1_{\star}\\) and \\(f\\).
The morphisms have to obey one of the following sets of equations:

\[
\begin{array}{c|cc}
\circ & 1_\star & f \\\\
\hline
1_\star & 1_\star & f \\\\
f & f & 1_\star
\end{array}
\]

or

\[
\begin{array}{c|cc}
\circ & 1_\star & f \\\\
\hline
1_\star & 1_\star & f \\\\
f & f & f
\end{array}
\]