Let $U$ be a set. Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation on $U$ (that is $f$ is a set of functions $N\rightarrow U$).

I denote $\mathcal{L}\in \upuparrows f \Leftrightarrow \forall L\in\prod_{i\in N}\mathcal{L}_i: f \cap \prod_{i\in N} L_i\ne\emptyset$ for every $N$-indexed family $\mathcal{L}$ of filters on $U$.

Does $\mathcal{L}\in \upuparrows f$ imply existence of $a\in \prod_{i\in N} \operatorname{atoms} \mathcal{L}_i$ (where $\operatorname{atoms}\mathcal{X}$ is the set of ultrafilters over filter $\mathcal{X}$) such that $\forall A\in\prod_{i\in N} a_i: f\cap \prod_{i\in N} A_i\ne\emptyset$?

This question is probably related with my other question.

Let $U$ be a set. Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation on $U$ (that is $f$ is a set of functions $N\rightarrow U$).

I denote $\mathcal{L}\in \upuparrows f \Leftrightarrow \forall L\in\prod_{i\in N}\mathcal{L}_i: f \cap \prod_{i\in N} L_i\ne\emptyset$ for every $N$-indexed family $\mathcal{L}$ of filters on $U$.

Does $\mathcal{L}\in \upuparrows f$ imply existence of $a\in \prod_{i\in N} \operatorname{atoms} \mathcal{L}_i$ (where $\operatorname{atoms}\mathcal{X}$ is the set of ultrafilters over filter $\mathcal{X}$) such that $\forall A\in\prod_{i\in N} a_i: f\cap \prod_{i\in N} A_i\ne\emptyset$?

This question is probably related with my other question.