Powerset functor on category Rel (sets and relations) Please will someone kindly explain how the powerset functor maps arrows in the category Rel. I understand that sets (objects) are mapped to their corresponding powerset, but I can't get my head around the arrows. If someone wouldn't mind giving a small example i'd be terribly grateful.
I found this explanation, but the notation confuses me:
 
Thank you
Martin

If $f:A\to B$ is a morphism in Rel, then it is relation between sets $A$ and $B, \ $ whose domain consists of certain elements $x$ of $A$, and whose codomain consists of certain elements $y$ of $B.$
Likewise,  $\mathscr Pf:\mathscr PA\to \mathscr PB$ is a relation whose domain consists of certain elements $a$ of $\mathscr P(A),\ $ and whose codomain consists of certain elements $b$ of $\mathscr P(B). $ 
According to the definition, $a\mathscr Pfb\ $ just in case there is an $x\in a$ and a $y\in b$ such that $xfy.$
One checks easily using the definition of composition of relations that $\mathscr P$ is a functor.
