John, I have a question about your hint in [#13](https://forum.azimuthproject.org/discussion/comment/18346/#Comment_18346).

You said that

> Conversely, if our monotone map between posets _does_ preserve all joins (resp. meets) then it _does_ have a left (resp. right) adjoint. But in this case it's often just as easy to guess the desired adjoint and prove it obeys the definition of adjoint.

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:

> By the way, if we want to prove a monotone function between _posets_ has no left (resp. right) adjoint, one option is to use the [Adjoint Functor Theorem for posets](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1) and show that it doesn't preserve all joins (resp. meets).

You said that

> Conversely, if our monotone map between posets _does_ preserve all joins (resp. meets) then it _does_ have a left (resp. right) adjoint. But in this case it's often just as easy to guess the desired adjoint and prove it obeys the definition of adjoint.

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:

> By the way, if we want to prove a monotone function between _posets_ has no left (resp. right) adjoint, one option is to use the [Adjoint Functor Theorem for posets](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1) and show that it doesn't preserve all joins (resp. meets).