Sophie wrote:

> In [#10](https://forum.azimuthproject.org/discussion/comment/18346/#Comment_18346) 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.