**Puzzle JMC4:** Find the right adjoint of \$$f : \mathbb{N} \to \mathbb{N}\$$ defined by \$$f(x) = \lfloor \frac{x}{2} \rfloor\$$.

**Puzzle JMC5:** Call the above right adjoint \$$g\$$. Why does \$$g\$$ _not_ have a right adjoint? (What would it be if it did, and how can we formalize this?)

These adjoints all make a nice pattern. It seems like the pattern would "never end" if we worked in \$$\mathbb{Z} \to \mathbb{Z}\$$, i.e. the chain of adjoints (in both directions) containing \$$x \mapsto 2x\$$ never terminates.

Also, it seems like even in \$$\mathbb{N} \to \mathbb{N}\$$, the sequence of left adjoints starting from \$$x \mapsto 2x\$$ never ends. However, I've computed a few of these out, and it seems to approach the identity function the further out you go.

(EDIT: JMC4 and JMC5 are just Exercise 1.80 in the book, hah. Oops.)