This conjecture appeared to be false.

Now I propose an alternative conjecture:

Let $latex A$, $latex B$ be sets.

**Conjecture** Funcoids $latex f$ from $latex A$ to $latex B$ bijectively corresponds to the sets $latex R$ of pairs

$latex (\mathcal{X}; \mathcal{Y})$ of filters (on $latex A$ and $latex B$ correspondingly) that

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

by the mutually inverse formulas:

$latex (\mathcal{X} ; \mathcal{Y}) \in R \Leftrightarrow \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \sqsubseteq f \quad \mathrm{and} \quad f = \bigsqcup^{\mathsf{FCD}} \left\{ \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\}$.