  We pursue the study of Ultracategories initiated by Makkai and more recently
Lurie by looking at properties of Ultracategories of complete metric
structures, i.e. coming from continuous model theory, instead of
ultracategories of models of first order theories. Our main result is that for
any continuous theory $\mathbb{T}$, there is an equivalence between the
category of left ultrafunctors from a compact Hausdorff space $X$ to the
category of $\mathbb{T}$-models and a notion of bundle of $\mathbb{T}$-models
over $X$. The notion of bundle of $\mathbb{T}$-models is new but recovers many
classical notions like Bundle of Banach spaces, or (semi)-continuous field of
$\mathrm{C}^*$-algebras or Hilbert spaces.
