I think for **Puzzle 18** we also need \$$f\$$ surjective? Since if we take \$$y \in Y \setminus f_{\ast}(X)\$$, then for a left adjoint \$$g_{\ast}\$$:

$$g_{\ast}(\\{y\\}) \subseteq X \iff \\{y\\} \subseteq f_{\ast}(X)$$

where the left hand side is clearly always true, but the right hand side is false by definition.