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.

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

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