This question was inspired by http://mathoverflow.net/questions/180282/can-we-build-a-continuous-function-from-fibers-preimages-defined-over-a-topolo

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 $U\in L$?