ScalingStacks

Proof. Since $\cat_{n}$ is locally finitely presentable, it suffices to show that the inclusion $\gaunt_{n}\hookrightarrow\cat_{n}$ commutes with filtered colimits. To this end, suppose $\Lambda$ a filtered category, and suppose $D\colon\Lambda\to\cat_{n}$ a diagram such that for any object $\alpha\in\Lambda$, the $n$-category $D_{\alpha}$ is gaunt. We claim that the colimit $D=\colim_{\alpha\in\Lambda}D_{\alpha}$ (formed in $\cat_{n}$) is gaunt as well. This claim now follows readily from the fact that both $C_{k}$ and $\sigma^{k}(E)$ are compact objects in $\cat_{n}$. ∎