If anyone here read these lectures but not my recent corrections, they may be worth another look:

The right adjoint to the function \$$f : \mathbb{N} \to \mathbb{N}\$$ that doubles any natural number is \$$g(n) = \lfloor n/2\rfloor \$$, not \$$g(n) = \lceil n/2\rceil \$$. With this fixed it becomes an example of one of my favorite principles:

> The left adjoint comes as close as possible to the (perhaps nonexistent) correct answer while making sure to _never choose a number that's too small_. The right adjoint comes as close as possible while making sure to _never choose a number that's too big_.

> The two adjoints represent two opposing philosophies of life: _make sure you never ask for too little_ and _make sure you never ask for too much_. If you need a mnemonic to remember which is which, remember left adjoints are "left-wing" or "liberal" or "generous", while right adjoints are "right-wing" or "conservative" or "cautious".

* [Lecture 9 - Chapter 1: Adjoints in the Logic of Subsets.](https://forum.azimuthproject.org/discussion/1931/lecture-9-adjoints-and-the-logic-of-subsets)

I changed my notation to match that in the book, and more importantly fixed a serious mistake. With all this fixed, we see that given any function \$$f : X \to Y\$$ between sets, the left adjoint of the **inverse image** \$$f^{*} : PY \to PX\$$ is \$$f_{!} : PX \to PY \$$, given by

$$f_{!}(S) = \\{y \in Y: x \in S \textrm{ for some } x \textrm{ such that } y = f(x)\\}$$

for any \$$S \subseteq X\$$, while the right adjoint is \$$f_{\ast} : PX \to PY \$$, given by

$$f_{\ast}(S) = \\{y \in Y: x \in S \textrm{ for all } x \textrm{ such that } y = f(x)\\} .$$

This lets us see yet another example of those principles. Let \$$X \$$ be the set of states of your room, and \$$Y \$$ the set of states of a thermometer in your room: that is, thermometer readings. Let \$$f : X \to Y \$$ map any state of your room to the thermometer reading. Then:

**Puzzle 24.** What is \$$f_{!}(\\{\text{there is a living cat in your room}\\})\$$? How is this an example of the "liberal" or "generous" nature of left adjoints, meaning that they're a "best approximation from above"?

**Puzzle 25.** What is \$$f_{\ast}(\\{\text{there is a living cat in your room}\\})\$$? How is this an example of the "conservative" or "cautious" nature of right adjoints, meaning that they're a "best approximation from below"?