For \$$f_\ast(\\{\text{there is a living cat in your room}\\})\$$, I get the set of all thermometer readings which are *only* possible if there is a live cat in the room. (This is interpreting the state set {there is a living cat in your room} not as a singleton but as the set of all room states in which the room contains a living cat.) I imagine for most interpretations of the function \$$f\$$ this set is empty.

If we think of \$$f_!\$$ and \$$f_\ast\$$ as best approximations to an imaginary "inverse" of \$$f^\ast\$$, then we're trying to find the best approximation from above or below to some imaginary set of temperature readings that correspond exactly to the room states containing a live cat. The "liberal" left adjoint \$$f_!\$$ generously throws in all the temperatures for which *some* corresponding room state contains a live cat, while the "conservative" right adjoint \$$f_\ast\$$ cautiously only allows the temperatures for which *all* corresponding room states contain a live cat.