For any set \$$X\$$ let \$$PX\$$ be the **[power set](https://en.wikipedia.org/wiki/Power_set)** of set, namely the set of all subsets of \$$X\$$. It's easy to see that the subset relation \$$\subseteq\$$ makes \$$PX\$$ into a poset. Suppose we have any function between sets

$$f : X \to Y$$

This gives a function

$$f_{!} : PX \to PY$$

sending each subset \$$S \subseteq X\$$ to its **[image](https://en.wikipedia.org/wiki/Image_(mathematics)#Image_of_a_subset)** under \$$f\$$:

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

**Puzzle 17.** Show that \$$f_{!} : PX \to PY \$$ is a monotone function.

**Puzzle 18.** Does \$$f_{!} \$$ always have a left adjoint? If so, describe it. If not, give an example where it doesn't, and some conditions under which it *does* have a left adjoint.

**Puzzle 19.** Does \$$f_{!}\$$ always have a right adjoint? If so, describe it. If not, give an example where it doesn't, and some conditions under which it *does* have a right adjoint.

It would be nice if experts hold back on answering these puzzles until others have had a crack at them. I asked them in a slightly odd way which, I hope, will make them more enlightening to solve.