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]\$$.