  We prove a conjecture of Emerton, Gee and Hellmann concerning the
overconvergence of \'etale $(\varphi,\Gamma)$-modules in families parametrized
by topologically finite type $\mathbb{Z}_{p}$-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.
