Classicality of derived Emerton-Gee stack

We construct a derived stack $X$ of Laurent F-crystals on , where $O_K$ is the ring of integers of a finite extension K of $Q_p$ . We first show that its underlying classical stack $_{}^{\text{cl}}X$ coincides with the Emerton-Gee stack $X_{\text{EG}}$ , i.e. the moduli stack of étale $(\phi ,\Gamma )$ -modules. Then we prove that the derived stack $X$ is classical in the sense that when restricted to truncated animated rings, $X$ is equivalent to the sheafification of the left Kan extension of $X_{\text{EG}}$ along the inclusion from the classical commutative rings to animated rings.