Let $X_{\bullet}^+$ be a strictly simplicial algebraic space and for a morphism $\delta:[m]\to[n]$ in $\Delta^+$, let $\delta:X_n\to X_m$ also denote the associated map (by abuse of notation). Then one can consider the category $\operatorname{Mod}(\mathcal{O}_{X_{\bullet}^+})$ of $\mathcal{O}_{X_{\bullet}^+}$-modules in the étale topos of $X_{\bullet}^+$. Let $D(\mathcal{O}_{X_{\bullet}^+})$ be the associated derived category. Let $D'$ be the category whose objects are pairs $(E_n,\varphi_{\delta})$, where for each $n$, $E_n$ is an object of $D(\mathcal{O}_{X_n})$, and for $\delta:[m]\to[n]$ we have a morphism $\varphi_{\delta}:\delta^*E_{m}\to E_{n}$, such that the various $\varphi_{\delta}$ are compatible with composition.

It appears that we have a morphism of (triangulated?) categories $D(\mathcal{O}_{X^+_{\bullet}})\to D'$ given by restriction. Is this morphism an equivalence?

I was trying to construct a quasi-inverse by replacing a system $(E_n,\varphi_{\delta})$ with an isomorphic one in which the $\varphi_{\delta}$ are given by actual complex maps, and the compatibility holds on the level of complexes. This way, one should get an object in $D(\mathcal{O}_{X^+_{\bullet}})$. However, I can only get the compatibility up to homotopy, so it seems like this doesn't work.