What are hyperfuncoids isomorphic to?

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

Products are \prod A for A \in \prod \mathfrak{A}.

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

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

  1. prestaroids on \mathfrak{A};
  2. staroids on \mathfrak{A};
  3. completary staroids on \mathfrak{A}?

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

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

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