Corollary 7.2. Assuming that:

$σ$ an $L$ -sentence
${(M_{j})}_{j \in J}$ a family of non-empty $L$ -structures
$U$ an ultrafilter on $J$
$M = \prod_{j \in J} M_{j} ∕ U$

Then

M ⊨ σ

if and only if

{j \in J : M_{j} ⊨ σ} \in U

.