Let $(X, \preceq)$ and $(Y, \ll)$ be partially ordered sets. Suppose there are functions $f : X \to Y$ and $g : Y \to X$ such that $$x \preceq g(y) \iff y \ll f(x)$$ for all $x \in X$, $y \in Y$.
Define the closed elements of $X$ to be all elements of the form $g(f(x))$ and the closed elements of $Y$ to be all elements of the form $f(g(y))$. I have to prove that there is an order-reversing bijection between the closed elements of $X$ and the closed elements of $Y$.
I'm really not sure how to do this. I'm thinking the map would have to be something simple but I can't see what it would be. If you just map $g(f(x)) \mapsto f(x)$ then that isn't necessarily injective and also maps onto $f(X)$ which might be larger than the set of closed elements of $Y$.
I feel like I have to use the property of $f$ and $g$ listed above to derive some other relationship between $f$ and $g$ ... maybe something like $f(g(f(x))) = f(x)$, but I'm not sure.