**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.)

**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.)