This question was inspired by Can we build a continuous function from "fibers"/preimages defined over a topological base?

Let $X,Y$ be sets and $L\subseteq \mathcal{P}(Y)$. Suppose $L$ has the following properties:

• $\emptyset, Y\in L$;
• $L$ is closed under arbitrary unions and intersections.

Let $F: L\to \mathcal{P}(X)$ be a function such that

• $F(\emptyset) = \emptyset$ and $F(Y) = X$;
• $F$ commutes with arbitrary unions and intersections.

Is there a function $f: X \to Y$ such that $f^{-1}(U) = F(U)$ for all $U\in L$?

This question was inspired by Can we build a continuous function from "fibers"/preimages defined over a topological base?

Let $X,Y$ be sets and $L\subseteq \mathcal{P}(Y)$. Suppose $L$ has the following properties:

• $\emptyset, Y\in L$;
• $L$ is closed under arbitrary unions and intersections.

Let $F: L\to \mathcal{P}(X)$ be a function such that

• $F(\emptyset) = \emptyset$ and $F(Y) = X$;
• $F$ commutes with arbitrary unions and intersections.

Is there a function $f: X \to Y$ such that $f^{-1}(U) = F(U)$ for all $U\in L$?