Let $latex \mathfrak{A}$ be an indexed family of sets.

*Products* are $latex \prod A$ for $latex A \in \prod \mathfrak{A}$.

*Hyperfuncoids* are filters $latex \mathfrak{F} \Gamma$ on the lattice $latex \Gamma$ of all finite unions of products.

**Problem**

Is $latex \bigsqcap^{\mathsf{FCD}}$ a bijection from hyperfuncoids $latex \mathfrak{F} \Gamma$ to:

- prestaroids on $latex \mathfrak{A}$;
- staroids on $latex \mathfrak{A}$;
- completary staroids on $latex \mathfrak{A}$?

If yes, is $latex \mathrm{up}^{\Gamma}$ defining the inverse bijection?

If not, characterize the image of the function $latex \bigsqcap^{\mathsf{FCD}}$ defined on $latex \mathfrak{F} \Gamma$.

Consider also the variant of this problem with the set $latex \Gamma$ replaced with the set $latex \Gamma^{\ast}$ of complements of elements of the set $latex \Gamma$.