  We develop the notion of coherent ultrafilters (extenders without normality
or well-foundedness). We then use definable coherent ultraproducts to
characterize any extension of a model $M$ in any fragment of
$\mathbb{L}_{\infty, \omega}$ that defines Skolem functions by a sufficiently
complete (but in $ZFC$) coherent ultrafilter. We apply this method to various
elementary classes and AECs.
