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Proposition 7.5.2.11. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories, and suppose that the category $\operatorname{\mathcal{C}}$ has an initial object. Then the comparison map $\iota : \varprojlim (\mathscr {F}) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ of Remark 7.5.2.10 is an equivalence of $\infty $-categories.

Proof. Let $C \in \operatorname{\mathcal{C}}$ be an initial object, so that the inclusion map $\{ C\} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is left anodyne (Corollary 4.6.6.25). Applying Remark 7.5.2.7, we see that evaluation at $C$ induces an equivalence of $\infty $-categories $\operatorname{ev}_{C}: \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \mathscr {F}(C)$. Our assumption that $C$ is initial also guarantees that the composition $(\operatorname{ev}_ C \circ \iota ): \varprojlim (\mathscr {F}) \rightarrow \mathscr {F}(C)$ is an isomorphism of simplicial sets, so that $\varprojlim (\mathscr {F})$ is an $\infty $-category and $\iota $ is an equivalence of $\infty $-categories. $\square$