Has anyone figured out an answer for **puzzle 85**?

Here's what I got so far.

The "would be" left adjoint should look like:

\[ h(x) = \bigwedge \\left\\{ y \in \mathbb{N}[S] \; : \; f(y) \to x \\right\\} . \]

I've tried instantiating the formula for a resource in \\( \mathbb{N}[T] \\) – I picked \\([\textrm{egg}]\\) and I've used the fact that \\( f \\) forgets bowls and shells:

\[ h([\textrm{egg}]) = \bigwedge \\left\\{ [\textrm{egg}], [\textrm{egg}] + [\textrm{bowl}], [\textrm{egg}] + [\textrm{shells}], [\textrm{egg}] + [\textrm{bowl}] + [\textrm{shells}], \cdots \\right\\} . \]

But I cannot find an element in \\( \mathbb{N}[S] \\) that is less than (that is, can be produced from) both \\( [\textrm{egg}] \\) and \\( [\textrm{egg}] + [\textrm{bowl}] \\).

I'm tempted to conclude there is no left adjoint, but I doubt my reasoning as it goes against [John's optimism](https://forum.azimuthproject.org/discussion/comment/18382/#Comment_18382).

Here's what I got so far.

The "would be" left adjoint should look like:

\[ h(x) = \bigwedge \\left\\{ y \in \mathbb{N}[S] \; : \; f(y) \to x \\right\\} . \]

I've tried instantiating the formula for a resource in \\( \mathbb{N}[T] \\) – I picked \\([\textrm{egg}]\\) and I've used the fact that \\( f \\) forgets bowls and shells:

\[ h([\textrm{egg}]) = \bigwedge \\left\\{ [\textrm{egg}], [\textrm{egg}] + [\textrm{bowl}], [\textrm{egg}] + [\textrm{shells}], [\textrm{egg}] + [\textrm{bowl}] + [\textrm{shells}], \cdots \\right\\} . \]

But I cannot find an element in \\( \mathbb{N}[S] \\) that is less than (that is, can be produced from) both \\( [\textrm{egg}] \\) and \\( [\textrm{egg}] + [\textrm{bowl}] \\).

I'm tempted to conclude there is no left adjoint, but I doubt my reasoning as it goes against [John's optimism](https://forum.azimuthproject.org/discussion/comment/18382/#Comment_18382).