Stoy, pages 117-122, discusses how to represent functions $f:\mathscr{P\omega\to P\omega}$ as elements $\mbox{graph}(f)\in\mathscr{P\omega}$, definition 7.5 page 120.
What would $\mbox{graph}(\mbox{id})\in\mathscr{P\omega}$ for the identity function be in this representation?
In the likely case you don't have Stoy's book handy, here are the relevant Pages 117-122. They'll introduce his notation, outline the procedure, etc, but assumes all necessary prerequisites...
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I should probably elaborate that after doing a little work on this myself (but not enough to find the actual answer, which seems a little trickier than I first imagined:), it became clear that the answer wasn't going to be what I'd hoped, that $\mbox{graph(id)}=\mathbb{N}$.
Towards the end of that section, on pages 121-122, Stoy mentions that various other representations are possible, and he gives the constraints that any such valid representation must satisfy. So my underlying/followup question here is: find a representation of $\mathscr{P\omega}\sim[\mathscr{P\omega\to P\omega}]$ such that $\mbox{graph(id)}=\mathbb{N}$. Or at least suggest how to go about finding such a representation, i.e., how would you set up the problem?