Kerodon

Corollary 4.5.6.17. Let $\operatorname{\mathcal{C}}$ be a small category, and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a levelwise categorical equivalence between isofibrant diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then the induced map $\varprojlim (\alpha ): \varprojlim ( \mathscr {F} ) \rightarrow \varprojlim ( \mathscr {G} )$ is an equivalence of $\infty $-categories.