Example 7.5.5.3 (Limits of Isofibrant Diagrams). Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ be a limit diagram in the category of simplicial sets. Suppose that the diagram $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$ is isofibrant (Definition 4.5.6.3). Then $\overline{\mathscr {F}}$ is a categorical limit diagram of $\infty $-categories: this follows by combining Corollary 4.5.6.13 with Proposition 7.5.3.12.
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