Order-Preserving Adjunction?

Question

Let there be two categories PrO (category of preordered sets) and ProM (category of preordered monoids). If we know that there exists an adjunction $F\dashv G$ for functors $F:\textbf{PrO}\to\textbf{ProM}$ and $G:\textbf{ProM}\to\textbf{PrO}$, how can we show that this adjunction is order-preserving?

Answer 1

We can first prove that PrO and ProM are in adjunction by saying that the natural isomorphism,

$$\text{Hom}_\textbf{ProM}(F(A),A^*)\cong\text{Hom}_\textbf{PrO}(A,G(A^*)),$$

exists iff there exists a bijection between these hom-sets and this isomorphism is natural in $A$ and in $A^*$.

Then we may try and proving that the adjunction $F\dashv G$ is a galois connection of preordered sets.

OR you can just prove the galois connection directly.