Drinfeld-Lau Descent over Fibered Categories

Let $\mathcal{X}$ be a category fibered in groupoids over a finite field ${\mathbb{F}}_q$, and let $k$ be an algebraically closed field containing ${\mathbb{F}}_q$. Denote by ${\varphi }_k\text{\hspace{0.17em}⁣}:{\mathcal{X}}_k\to {\mathcal{X}}_k$ the arithmetic Frobenius of ${\mathcal{X}}_k\mathrm{/}k$ and suppose that $\mathcal{M}$ is a stack over ${\mathbb{F}}_q$ (not necessarily in groupoids). Then there is a natural functor $\alpha_{{\mathcal M},{\mathcal X}}\colon{\mathcal M}({\mathcal X})\to{\mathcal M}({\mathbf D_k}({\mathcal X}))$,where${\mathcal M}({\mathbf D_k}({\mathcal X}))$isthecategoryof$\phi_k$-invariantmaps${\mathcal X}k\to {\mathcal M}$\mathrm{.}AversionofDrinfel{d}^{\mathrm{\prime }}slemmastatesthatif${\mathcal X}$isaprojectiveschemeand${\mathcal M}$ is the stack of quasi-coherent sheaves of finite presentation, then ${\alpha }_{\mathcal{M},\mathcal{X}}$ is an equivalence. We extend this result in several directions. For proper algebraic stacks or affine gerbes $\mathcal{X}$, we prove Drinfeld’s lemma and deduce that ${\alpha }_{\mathcal{M},\mathcal{X}}$ is an equivalence for very general algebraic stacks $\mathcal{M}$. For arbitrary $\mathcal{X}$, we show that $\alpha{{\mathcal M},{\mathcal X}}$isanequivalencewhen${\mathcal M}$ is the stack of immersions, the stack of quasi-compact separated 'etale morphisms or any quasi-separated Deligne-Mumford stack with separated diagonal.