> **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}

\]

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}

\]