Puzzle 10: If \$$A, \leq_A\$$ is the ancestor graph (assuming everyone is a trivial self-ancestor) and \$$\mathbb{N}, \geq\$$ is the usual preorder given by the conventional interpretation of \$$\geq\$$ on the natural numbers, then the age function taking each person to their age in years is a monotone map: if \$$x\$$ is an ancestor of \$$y\$$, then \$$\textrm{age}(x) \geq \textrm{age}(y)\$$. (Note that the converse does not hold.)

There is also no requirement that \$$A, \leq_A\$$ and \$$B, \leq_B\$$ be distinct, so we could consider monotone maps from \$$A, \leq_A\$$ to itself. For example, let \$$A, \leq_A\$$ be the ancestry preorder in a colony of bacteria, and consider the mother function which maps each cell \$$x\$$ to its mother. (You might wonder why not its "parent", but it's an established convention in biology.) If \$$x\$$ is an ancestor of \$$y\$$, then \$$x\$$'s mother is an ancestor of \$$y\$$'s mother.

Puzzle 11: I have a question about this one. Suppose \$$A = \\{a, b, c, d\\}\$$ and \$$\leq_A\$$ is given by: \$$\leq_A = \\{(a, c), (b, d), (a, a), (b, b), (c, c), (d, d)\\}\$$. Let \$$(B, \leq_B)\$$ be given by \$$B=\\{1, 2, 3, 4\\}\$$ with \$$\leq_B\$$ having the usual interpretation. Finally define \$$f = \\{(a, 1), (b, 2), (c, 3), (d, 4)\\}\$$. Now \$$f\$$ is a monotone map with an inverse \$$g\$$, but \$$g\$$ isn't a monotone map. In particular, \$$2 \leq_B 3\$$ but we don't have \$$b \leq_A c\$$. Maybe I'm confused?

Puzzle 12: Let \$$g = \lceil\frac{n}{2}\rceil\$$.

Puzzle 13: Let \$$g = \lfloor{\frac{n}{2}}\rfloor\$$. The proofs in both cases are pretty direct.

Crudely speaking, if you draw \$$f\$$ and \$$g\$$ out on two natural number lines, then \$$f\$$ and \$$g\$$ being adjoints seems to amount to the condition that there are no "wires crossed" in the diagram. So if for every \$$n \in \mathbb{N}\$$ there are two \$$m\$$'s that \$$g\$$ could send it to, is the number of adjoints \$$2^\mathbb{N}\$$?

Apologies in advance for typos and thinkos.