Consider the property of the room "everyone in the room thinks it's too hot"; call it H. Then f_*(H) = {y in Y : for all states x of the room, if f(x)=y then x is in H}. That's a nice set: it's the set of temperatures that guarantee that everyone will think it's too hot.

But the property C "there's a cat in the room" would have f_*(C)=\emptyset because there's no temperature that guarantees there'll be a cat in the room.