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Proposition Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be an isofibrant diagram of $\infty $-categories. Then the inclusion map $\iota : \varprojlim ( \mathscr {F} ) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ is an equivalence of $\infty $-categories.

Proof. Let $\alpha : \mathscr {F} \hookrightarrow \mathscr {F}^{+}$ be the isofibrant replacement of Construction By virtue of Proposition (and Remark, it will suffice to show that the limit $\varprojlim (\alpha ): \varprojlim ( \mathscr {F} ) \hookrightarrow \varprojlim ( \mathscr {F}^{+} )$ is an equivalence of $\infty $-categories. This is a special case of Corollary, since $\alpha $ is a levelwise categorical equivalence between isofibrant diagrams (Proposition $\square$