$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.1.10.8. 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 satisfying the following conditions:
- $(1)$
For each $J \in \operatorname{\mathcal{J}}$, the $\infty $-category $\mathscr {F}(J)$ admits $K$-indexed colimits.
- $(2)$
For each morphism $u: J \rightarrow J'$ in $\operatorname{\mathcal{J}}$, the functor $\mathscr {F}(u): \mathscr {F}(J) \rightarrow \mathscr {F}(J')$ preserves $K$-indexed colimits.
Then the colimit $\varinjlim (\mathscr {F})$ admits $K$-indexed colimits. Moreover, a functor of $\infty $-categories $T: \varinjlim (\mathscr {F}) \rightarrow \operatorname{\mathcal{D}}$ preserves $K$-indexed colimits if and only if, for each $J \in \operatorname{\mathcal{J}}$, the composite functor $T_{J}: \mathscr {F}(J) \rightarrow \varinjlim (\mathscr {F}) \rightarrow \operatorname{\mathcal{D}}$ preserves $K$-indexed colimits.
Proof.
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
\[ (A, \leq ) \rightarrow \operatorname{QCat}\subset \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto \operatorname{\mathcal{C}}_{\alpha }. \]
Set $\operatorname{\mathcal{C}}= \varinjlim _{\alpha } \operatorname{\mathcal{C}}_{\alpha }$, where the colimit is formed in the category of simplicial sets. Since $(A, \leq )$ is $\kappa $-directed and $K$ is $\kappa $-small, every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ can be lifted to a diagram $f_{\alpha }: K \rightarrow \operatorname{\mathcal{C}}_{\alpha }$ for some $\alpha \in A$ (see Variant 9.2.2.10). Using assumption $(1)$, we see that $f_{\alpha }$ can be extended to a colimit diagram $\overline{f}_{\alpha }: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\alpha }$. Using assumption $(2)$ and Proposition 9.1.10.5, we see that the image of $\overline{f}_{\alpha }$ in $\operatorname{\mathcal{C}}$ is a colimit diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ extending $f$. This proves the first assertion. Moreover, it shows that every colimit diagram in $\operatorname{\mathcal{C}}$ can be obtained (up to isomorphism) from a colimit diagram in some $\operatorname{\mathcal{C}}_{\alpha }$. Consequently, if $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories having the property that $T|_{\operatorname{\mathcal{C}}_{\alpha }}$ preserves $K$-indexed colimits for each $\alpha \in A$, then the functor $T$ itself preserves $K$-indexed colimits. The converse follows from Proposition 9.1.10.5.
$\square$