Sophie wrote:

> In [#10]( I suggested a reason why \\(f\\) does not have a right adjoint but I'm still wondering if I made an error. I think I'll try tackling this from the direction John suggested:

I think you're correct about \\(f\\) not having a right adjoint. If \\(f\\) has a right adjoint \\(g\\) then it's given by the formula in my last comment. And this formula can be translated into the following more intuitive description:

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

iff \\(x \in \mathbb{N}[S] \\),is biggest combination of bowls, eggs, yolks, whites and egg shells 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]\\).

However, there _is no_ biggest combination with this property! Because \\(f\\) ignores all bowls and egg shells, if \\(x\\) obeys the above inequality so does \\(x + [\textrm{bowl}] \\).

Does this argument sound valid? Don't trust me: I often get things backwards.

However, this makes me more optimistic that \\(f\\) has a _left_ adjoint.