  We construct a derived stack $\chi$ of Laurent $F$-crystals on
$(\mathcal{O}_K)_{\mathbb{\Delta}}$, where $\mathcal{O}_K$ is the ring of
integers of a finite extension $K$ of $\mathcal{Q}_p$. We first show that its
underlying classical stack $^{\rm cl}\chi$ coincides with the Emerton-Gee stack
$\chi_{\rm EG}$, i.e., the moduli stack of \'etale $(\phi, \Gamma)$-modules.
Then we prove that this derived stack is classical in the sense that when
restricted to truncated animated rings, $\chi$ is equivalent to the
sheafification of the left Kan extension of $\chi_{\rm EG}$ along the inclusion
from the classical commutative rings to animated rings.
