Regarding Puzzle 21, if we consider \\(\mathbb{N} = \\{0, 1, 2, \ldots\\}\\), then we can construct a left-adjoint function \\(f : \mathbb{R}\to\mathbb{N}\\) to our \\(i : \mathbb{N}\to\mathbb{R}\\) such that

$$f = \begin{cases}
0 & \text{if } x\le0\\\\
\lceil x \rceil & \text{if } x>0
\end{cases}$$

We see that our function \\(f\\) is monotone since if \\(a \le b\\) then \\(f(a) \le f(b)\\). Then, we can check that if \\(x \le i(y)\\), then \\(f(x) \le y\\), which is the definition of a left-adjoint.