Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

We study several properties and applications of the ultrapower MU a metric space M. prove that Lipschitz-free F(MU) is finitely representable in F(M). also characterize spaces are Lipschitz Banach as those biLipschitz embed into an space. Thanks to this link, we obtain if M X, then F(M) F(X). apply these results cotype stability functions under ultraproducts.