= 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 condition
$$
\boxed{f(x)\leq y\iff x\leq g(y).}
$$
The <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 $x\leq gf(x)$ and $fg(y)\leq y$. These inequalities imply the displayed equivalence: $f(x)\leq y$ gives $x\leq gf(x)\leq g(y)$, and $x\leq g(y)$ gives $f(x)\leq fg(y)\leq y$.
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