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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

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    It's better to embed a picture for readers' convenience. – 2017-02-07
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    Hi, I couldn't embed the image as it said I needed 10 'reputation' to post images. Apologies. – 2017-02-07
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    It's me to apologize since I was unaware of that. Now you may do so. – 2017-02-08
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    Thank you, i've amended my post accordingly. – 2017-02-08

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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.

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    Thank you very much, I think I was reading too much into the definition as this seems very straight forward :) – 2017-02-08