Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 5 / a / Solution

Since the hom-sets in these categories are either empty or singleton, an adjunction $f⊣g$ is precisely the conditionThe hom-set bijections are then the unique possible ones, and naturality is automatic. This is a Galois connection between the two posets.

Equivalently, the monotone functions satisfy the unit and counit inequalities $x\le gf(x)$ and $fg(y)\le y$. The equivalence implies these by taking $y=f(x)$ and $x=g(y)$. Conversely, if the inequalities hold, $f(x)\le y$ implies $x\le gf(x)\le g(y)$, while $x\le g(y)$ implies $f(x)\le fg(y)\le y$.