Yes, David just answered Puzzle 19. The **preimage** or **inverse image** as David just defined it:

$$f^{\ast}(S) = \\{x \in X: f(x) \in S\\}$$

gives a monotone function \$$f^{\ast}: PY \rightarrow PX\$$ that is the right adjoint of \$$f_\{\ast} : PX \to PY \$$.

**Puzzle 20.** Does \$$f^{\ast}: PY \rightarrow PX \$$ have a right adjoint of its own?