If this puzzle is still intimidating, think a bit about what the right adjoint \\(g : \mathbb{N}[T] \to \mathbb{N}[S] \\) of \\(f : \mathbb{N}[S] \to \mathbb{N}[T] \\) would be like. Roughly speaking,

\[ g( b [\textrm{egg}] + c[\textrm{yolk}\ + d[\textrm{white}] ) \]

would be the biggest combination of bowls, eggs, yolks, whites and egg shells, say \\(x \in \mathbb{N}[S] \\), such that

\[ f(x) \le b [\textrm{egg}] + c[\textrm{yolk}\ + d[\textrm{white}] . \]

Here "biggest" means with respect to the preorder on \\(x \in \mathbb{N}[T]\\).

\[ g( b [\textrm{egg}] + c[\textrm{yolk}\ + d[\textrm{white}] ) \]

would be the biggest combination of bowls, eggs, yolks, whites and egg shells, say \\(x \in \mathbb{N}[S] \\), such that

\[ f(x) \le b [\textrm{egg}] + c[\textrm{yolk}\ + d[\textrm{white}] . \]

Here "biggest" means with respect to the preorder on \\(x \in \mathbb{N}[T]\\).