Proposition 9.1.10.5. Let $\kappa $ be a regular cardinal, let $K$ be a $\kappa $-small simplicial set, and let $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}$ be a $\kappa $-filtered diagram of $\infty $-categories. Assume that, for every morphism $u: J \rightarrow J'$ of $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}(u): \mathscr {F}(J) \rightarrow \mathscr {F}(J')$ preserves $K$-indexed colimits. Then, for every object $J \in \operatorname{\mathcal{J}}$, the natural map $\mathscr {F}(J) \rightarrow \varinjlim (\mathscr {F})$ preserves $K$-indexed colimits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Proposition 9.1.10.5. Let $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}$ be a $\kappa $-filtered diagram of $\infty $-categories, let $K$ be a $\kappa $-small simplicial set, and suppose that each of the transition functors $\mathscr {F}(J) \rightarrow \mathscr {F}(J')$ preserves $K$-indexed colimits. As in the proof of Proposition 9.1.10.1, we can reduce to the case where $\operatorname{\mathcal{J}}= \operatorname{N}_{\bullet }(A)$ is the nerve of a $\kappa $-directed partially ordered set $(A, \leq )$ and $\mathscr {F}$ is obtained from a strictly commutative diagram of simplicial sets
\[ (A, \leq ) \rightarrow \operatorname{QCat}\subset \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto \operatorname{\mathcal{C}}_{\alpha }. \]
In this case, we are reduced to showing that each of the functors $\operatorname{\mathcal{C}}_{\beta } \rightarrow \varinjlim _{\alpha \in A} \operatorname{\mathcal{C}}_{\alpha }$ preserves $K$-indexed colimits (where the colimit is formed in the ordinary category of simplicial sets). Replacing $A$ by the subset $A_{\geq \beta } = \{ \alpha \in A: \alpha \geq \beta \} $, we can reduce to the case where $\beta $ is the least element of $A$. In this case, the result follows from Lemma 9.1.10.7. $\square$