ScalingStacks

Proof. Let $X\in\precat_{(\infty,n)}$ be an object, and consider the map $RX\to X$. The claim is that $j^{*}RX\to j^{*}X$ is an equivalence. Since the cells generate $S^{-1}\pre(\Upsilon_{n})$ under colimits, it’s enough to observe that $\map(C_{i},RX)\simeq(RX)(C_{i})\to X(C_{i})\simeq Map(C_{i},X)$ is an equivalence, for any cell $C_{i}$. ∎