> PZ100- my guess..16 morphisms and yes, its a preorder.

16 morphisms seems too small IMO.

Here are all of the objects in the category:

\$\newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{|c|c|c|c|} \hline \ \ ^◤& \ \ _◥& _◢\ \ & ^◣\ \ \\\\ \hline \ \ _◣ & _◤\ \ & ^◥\ \ & \ \ ^◢ \\\\ \hline \end{array} \$

For any pair of objects, there are at least two morphisms.

For instance, take \$$\boxed{\ \ ^◤}\$$ and \$$\boxed{^◣ \ \ }\$$. We can get from the first diagram to the second with two reflections:

\$\begin{eqnarray} \boxed{\ \ ^◤} & \underset{\text{vertical reflection}}{\longrightarrow} & \boxed{^◥\ \ } \\\\ \boxed{^◥\ \ } & \underset{\text{NW-SE diagonal reflection}}{\longrightarrow} & \boxed{^◣ \ \ } \end{eqnarray} \$

We can get back by doing the same two reflections in reverse order.

Since there are are also identity morphisms, **there must be at least 64 morphisms**.

There are multiple transformations to get from \$$\boxed{\ \ ^◤}\$$ to \$$\boxed{^◣ \ \ }\$$, however. I don't know if John wants to count these as the same...