ScalingStacks

Remark 2.8. Suppose we are given a spectral functor $F\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$. Since $\widehat{{\mathcal{A}}}^{\cf}$ is generated by $\widehat{{\mathcal{A}}}_{\perf}$ under filtered homotopy colimits and Morita equivalences are stable under filtered homotopy colimits, it follows that $F$ is a Morita equivalence if and only if $F_{!}^{\cf}\colon\widehat{{\mathcal{A}}}^{\cf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{{\mathcal{B}}}^{\cf}$ is a DK-equivalence.