Funcoids are filters? Conjecture II

Earlier I have conjectured that the set of funcoids is order-isomorphic to the set of filters on the set of finite joins of funcoidal products of two principal filters. For an equivalent open problem I found a counterexample.

Now I propose another similar but weaker open problem:

Conjecture Let $latex U$ be a set. The set of funcoids on $latex U$ is order-isomorphic to the set of filters on the set $latex \Gamma$ (moreover the isomorphism is (possibly infinite) meet of the filter), where $latex \Gamma$ is the set of unions $latex \bigcup_{X\in S}(X\times Y_X)$ where $latex S$ is a finite partition of $latex U$ and $latex Y\in \mathscr{P} U$ for every $latex X\in S$

The last conjecture is equivalent to this question formulated in elementary terms. If you solve this (elementary) problem, it could be a major advance in mathematics.