Exactly. (I didn't realize the comments were continuing down here and made the essentially same point here after Phil amended the post.)

The interesting point that Phil raises is whether there's any reason to have a particular risk preference with respect to u. I'm not sure that the analogy between being inequality averse amongst possible "me"s and and inequality averse amongst actual others gets much traction once we remember that probability is in the mind. But it's an interesting question nonetheless.

Allais, in particular argued that any form of risk preference over u should be allowable, and Broome finds this view "very plausible". All of which seems to make rational decision-making under uncertainty much more difficult, particularly as it's far from obvious that we have intuitive access to these risk preferences. (I certainly don't have intuitive access to mine.)

P.S. I assume you mean f(u)-maximizer rather than f''(u)-maximizer?