But \$$f: 2^X \rightarrow 2^Y\$$ and \$$g: 2^Y \rightarrow 2^X\$$ are not just any pair of order-reversing functions, they have an additional property which classifies them as a **Galois connection**:

\$f(x) \supseteq y \iff x \subseteq g(y)\$

Using Willerton's language, these are equivalent because they both state that the objects in \$$x\$$ have all the attributes in \$$y\$$.