Looking back, it seems like that direction of the adjoint functor theorem is conditioned on \$$\mathbb N [S]\$$ having all joins (resp. meets). But I think \$$\mathbb N [S]\$$ has neither since \$$x \leq x'\$$ means that \$$x\$$ and \$$x'\$$ must have the same number of bowls. I gave some reasons why in [#10](https://forum.azimuthproject.org/discussion/comment/18346/#Comment_18346). So this direction of the adjoint functor theorem won't apply to our specific puzzle. Is this correct?
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: