What are hyperfuncoids isomorphic to?

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:

1. prestaroids on $latex \mathfrak{A}$;
2. staroids on $latex \mathfrak{A}$;
3. 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 $latex \mathfrak{F} \Gamma$.