Given a partial ordered set $P$ we can define a category $\cal P$ with $\text{Ob}({\cal{P}})=P$ and for $A,B\in P$ $${\cal P}(A,B)=\begin{cases}\{\cdot\}&\text{if }A\ge B,\\\emptyset&\text{else}.\end{cases}.$$
$\{\cdot\}$ is a single element set.
I don't quite understand this category. Does $\{\cdot\}$ depends on $A,B$? What is the composition map $\{\cdot\}\circ\{\cdot\}$ or $\{\cdot\}\circ\emptyset?$
Thank you!
For each pair of objects A and B, the hom-set ${\cal{P}}(A,B)$ has exactly one arrow if $A\geq B$ and no arrow otherwise. This is by definition. If $A\geq B$ and $B\geq C$ then by transitivity $A\geq C$ so ${\cal{P}}(A,C)$ has an arrow. The composite of the arrow in ${\cal{P}}(A,B)$ and the arrow in ${\cal{P}}(B,C)$ must necessarily be the arrow in ${\cal{P}}(A,C)$. Associativity and the existence of an arrow in ${\cal{P}}(A,A)$ for every object $A$ is then easy to check. You may have been confused by the fact that your source used the same symbol for every arrow. It would be clearer to call the arrow in ${\cal{P}}(A,B)$ something like "$AgB$".