> 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...