Let $A$, $B$ be fixed sets.

What "means" the formula $Y \cap \alpha X \neq \emptyset \Leftrightarrow X \cap \beta Y \neq \emptyset$ for functions $\alpha:\mathscr{P}A\rightarrow\mathscr{P}B$ and $\beta:\mathscr{P}B\rightarrow\mathscr{P}A$? That is, how such pairs $(\alpha;\beta)$ can be represented?

Particularly, do there necessarily exist binary relations $p$ and $q$ such that $\alpha X=p[X]$ and $\beta Y=q[Y]$ for every $X\in\mathscr{P}A$, $Y\in\mathscr{P}B$?

By definition $y\in p[X] \Leftrightarrow \exists x\in X: (x,y)\in p$.

Such binary relations $p$ and $q$ exist. The proof:

$Y \cap \alpha X \neq \emptyset \Leftrightarrow X \cap \beta Y \neq \emptyset$;

$y \in \alpha X \Leftrightarrow X \cap \beta \{ y \} \neq \emptyset$;

$y \in \alpha X \Leftrightarrow \exists x \in X : x \in \beta \{ y \}$;

$y \in \alpha X \Leftrightarrow \exists x \in X : (x, y) \in p \Leftrightarrow y \in p [X]$ where $(x, y) \in p \Leftrightarrow x \in \beta \{ y \}$.

So $\alpha X = p [X]$. Similarly $\beta Y = q [Y]$ for a binary relation $q$ ($(y ; x) \in q \Leftrightarrow y \in \alpha \{ x \}$).

Also $(x, y) \in p \Leftrightarrow x \in \beta \{ y \} \Leftrightarrow \{ x \} \cap \beta \{ y \} \neq \emptyset \Leftrightarrow \{ y \} \cap \alpha \{ x \} \neq \emptyset \Leftrightarrow y \in \alpha \{ x \} \Leftrightarrow (y, x) \in q$. So $q = p^{- 1}$.

Also the formula $Y \cap \alpha X \neq \emptyset \Leftrightarrow X \cap \beta Y \neq \emptyset$ holds if $\alpha X=p[X]$ and $\beta Y=q[Y]$.

So this pair $(\alpha,\beta)$ is essentially just a binary relation $p=q^{-1}$.