Owen answered Puzzle 26 as follows:

> 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.

Good - yes, that's the answer I was fishing for! It's probably the empty set, unless for example we're only considering a restricted set of states where you own a cat who always comes inside when the temperature drops below freezing... not all physically possible states of the matter in your room.

> 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.

Exactly!!!

We're trying to figure out what the presence of a living cat in the room says about the temperature. The left and right adjoint give two ways of doing it: the generous way ("these are the temperatures where there _could_ be a live cat in the room") and the cautious way ("these are the temperature where there _must_ be one").

> 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.

Good - yes, that's the answer I was fishing for! It's probably the empty set, unless for example we're only considering a restricted set of states where you own a cat who always comes inside when the temperature drops below freezing... not all physically possible states of the matter in your room.

> 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.

Exactly!!!

We're trying to figure out what the presence of a living cat in the room says about the temperature. The left and right adjoint give two ways of doing it: the generous way ("these are the temperatures where there _could_ be a live cat in the room") and the cautious way ("these are the temperature where there _must_ be one").