I was trying to construct some element with specific properties in an ultraproduct and it boils down to a question which seems relatively natural but leaves me perfectly clueless.

$\textbf{Question:}$ Pick some $\alpha \in (0,1)$. Let $\phi:\mathcal{P} _{fin}(\mathbb{N}) \to \mathcal{P} _{fin}(\mathbb{N})$ be any function between finite sets which does not reduce cardinality too much, $\textit{i.e.}$ such that $|\phi(S)| \geq \alpha |S|$ and $\phi(S) \subset S$. Does there exists a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$ and a sequence of finite sets $S_i \subset \mathbb{N}$ such that $\forall s \in \mathbb{N}, \lbrace i \in \mathbb{N} \mid s \in \phi(S_i) \rbrace =: F_s \in \mathcal{U}$.

As far as I know, the value of $\alpha$ may be irrelevant, but this does not help me much: if $\alpha =1$ then the answer is obviously "yes" ($\phi$ must be the identity, so any increasing sequence of sets will do) and if $\alpha=0$ it's obviously "no" ($\phi$ may send every set to the empty set).

Also, does the requirement that $\phi$ is increasing ($\textit{i.e. } S \subset S' \Rightarrow \phi(S) \subset \phi(S')$ change anything?