Overconvergence of étale $(\varphi,\Gamma)$-modules in families

Abstract We prove a conjecture of Emerton, Gee and Hellmann concerning the overconvergence of étale $(\varphi,\Gamma)$ -modules in families parametrized by topologically finite-type $\mathbf{Z}_{p}$ -algebras. As a consequence, we deduce the existence of a natural map from the rigid fiber of the Emerton–Gee stack to the rigid analytic stack of $(\varphi,\Gamma)$ -modules.