On the $\infty$-stack of complexes over a scheme

We study fppf descent for enhanced derived categories. We revisit the work of [HS] and [TV08] in a lax context. More precisely, we construct a Cartesian and coCartesian fibration ${}^{\mathrm{op}}\mathscr D^+_S\rightarrow N(\mathop{\mathrm{Sch}}_S)$ whose fibre over an $S$-scheme $T$ is the opposite $\mathscr D^+(T)^{\mathrm{op}}$ of the quasicategory of bounded below complexes of $\mathscr O_T$-modules. We show that this fibration satisfies fppf-descent for schemes.