Let $(X_i, d_i, e_i)$ be a sequence of pointed metric spaces, let $\prod _\omega (X_i, d_i, e_i)$ be the ultraproduct of said spaces with respect to a nonprincipal ultrafilter $\omega$, and let $(\hat{X}, \hat{d})$ be the nonstandard hull of $\prod _\omega (X_i, d_i, e_i)$ (namely, the quotient under infinitesimal equivalence of the nearstandard subset of $\prod _\omega (X_i, d_i, e_i)$). Of course, $(\hat{X},\hat{d})$ is a bona fide standard metric space, so in particular we can consider the nonstandard extension/ultrapower $(^*\hat{X},^*\hat{d})$.
My question is: are $(^*\hat{X},^*\hat{d})$ and $\prod _\omega (X_i, d_i, e_i)$ isomorphic?