An important conjecture about funcoids. Version 2

This conjecture appeared to be false.

Now I propose an alternative conjecture:

Let A, B be sets.

Conjecture Funcoids f from A to B bijectively corresponds to the sets R of pairs
(\mathcal{X}; \mathcal{Y}) of filters (on A and B correspondingly) that

  1. R is nonempty.
  2. R is a lower set.
  3. every \left\{ \mathcal{X} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\} is a dcpo for every \mathcal{Y} \in \mathfrak{F}B and every \left\{ \mathcal{Y} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\} is a dcpo for every \mathcal{X} \in \mathfrak{F}A

by the mutually inverse formulas:
(\mathcal{X} ; \mathcal{Y}) \in R \Leftrightarrow \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \sqsubseteq f \quad \mathrm{and} \quad f = \bigsqcup^{\mathsf{FCD}} \left\{ \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\}.

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