Definable Coherent Ultrapowers and Elementary Extensions

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.