Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 20 / 7 / i / Solution

Each hom-set in a poset is either empty or a singleton. Thus a natural bijection $Y(fx,y)\cong X(x,gy)$ is precisely the conditionThe naturality squares commute automatically, because maps between the relevant singleton hom-sets are unique. This is a Galois connection. Equivalently, the adjunction unit and adjunction counit are the inequalities $x\le gf(x)$ and $fg(y)\le y$. These inequalities imply the displayed equivalence: $f(x)\le y$ gives $x\le gf(x)\le g(y)$, and $x\le g(y)$ gives $f(x)\le fg(y)\le y$.