> **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

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