Matthew wrote

>This argument I just gave to prove there is no companion used two things: \\(\hat{F} = \check{G} \iff F \dashv G\\) and the [Adjoint functor theorem for posets]( I did not explicitly invoke the adjoint functor theorem, however, just thought about it.

So that's how you use the Adjoint Functor Theorem. How useful! It is a tool to use when determining the existence of adjoints.

>The adjoint functor theorem says that the \\(G\\) you gave, if it is a right adjoint, must preserve all arbitrary meets. Can you see a meet it doesn't preserve?

While going through this I realized my understanding of the Adjoint Functor Theorem was very weak. I can see how the repaired \\(F(x)\\) doesn't preserve arbitrary joins since \\(4 \leq x \leq 6\\) and \\(6 \leq x\\) both map to 2. But for \\(G\\) I am not sure but maybe it's not preserving the top element which is \\(\infty\\) and instead setting 6 as the top element?