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.