Classicality of derived Emerton-Gee stack

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 ${}^{\mathrm{c}\mathrm{l}}\chi$ coincides with the Emerton-Gee stack ${\chi }_{\mathrm{E}\mathrm{G}}$, i.e., the moduli stack of étale $(\phi ,\mathrm{\Gamma })$-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 ${\chi }_{\mathrm{E}\mathrm{G}}$ along the inclusion from the classical commutative rings to animated rings.