Proposition 7.4.4.15. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram of $\infty $-categories having a limit $\varprojlim (\mathscr {F})$, let $K$ be a simplicial set, and suppose we are given a diagram $\overline{q}: K^{\triangleright } \rightarrow \varprojlim ( \mathscr {F} )$. For each vertex $C \in \operatorname{\mathcal{C}}$, let $\overline{q}_{C}$ denote the composition of $\overline{q}$ with the natural map $\varprojlim (\mathscr {F}) \rightarrow \mathscr {F}(C)$. If each $\overline{q}_{C}$ is a colimit diagram in the $\infty $-category $\mathscr {F}(C)$, then $\overline{q}$ is a colimit diagram in the $\infty $-category $\varprojlim ( \mathscr {F})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration having covariant transport representation $\mathscr {F}$ (for example, we can take $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$). Using Proposition 7.4.4.1, we can identify $\varprojlim ( \mathscr {F} )$ with the $\infty $-category $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian sections of $U$. For each object $C \in \operatorname{\mathcal{C}}$, our assumption on $\overline{q}_{C}$ guarantees that the evaluation map $\operatorname{ev}_{C}: \operatorname{Fun}_{/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_{C}$ carries $\overline{q}$ to a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ (see Remark 7.4.4.10), which remains a colimit diagram after applying the covariant transport functor $f_!: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ associated to any edge $f: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$. Applying Corollary 7.1.10.3, we deduce that $\overline{q}$ is a colimit diagram in the $\infty $-category of sections $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, and therefore also in the full subcategory $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. $\square$