Corollary 7.5.8.7. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{ \mathscr {F} }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a colimit diagram in the category of simplicial sets. If the diagram $\mathscr {F} = \overline{ \mathscr {F} }|_{ \operatorname{\mathcal{C}}}$ is projectively cofibrant, then $\overline{\mathscr {F}}$ is a categorical colimit diagram.
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Proof. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and define $\overline{\mathscr {G}}: (\operatorname{\mathcal{C}}^{\triangleright })^{\operatorname{op}} \rightarrow \operatorname{QCat}$ by the formula $\overline{\mathscr {G}}(C) = \operatorname{Fun}( \overline{\mathscr {F}}(C), \operatorname{\mathcal{D}})$. By virtue of Proposition 7.5.8.4, it will suffice to show that the diagram of Kan complexes $\overline{\mathscr {G}}^{\simeq }$ is a homotopy limit diagram. Setting $\mathscr {G} = \overline{\mathscr {G}}|_{ \operatorname{\mathcal{C}}^{\operatorname{op}} }$, our assumption that $\mathscr {F}$ is projectively cofibrant guarantees that the diagram $\mathscr {G}$ is isofibrant (Remark 7.5.6.6). It follows that the diagram of Kan complexes $\mathscr {G}^{\simeq }$ is also isofibrant, and that $\overline{\mathscr {G}}^{\simeq }$ is a limit diagram (Corollary 4.5.6.21). The desired result now follows from Example 7.5.4.2. $\square$