Lemma 9.1.10.7. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be the colimit of a diagram of simplicial sets $\{ \operatorname{\mathcal{C}}_{\alpha } \} _{\alpha \in A}$ indexed by a $\kappa $-directed partially ordered set $(A, \leq )$. Let $K$ be a $\kappa $-small simplicial set and let $\overline{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets which is obtained as the colimit of a compatible system of morphisms $\overline{F}_{\alpha }: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\alpha }$. Assume that each $\operatorname{\mathcal{C}}_{\alpha }$ is an $\infty $-category and that each $\overline{F}_{\alpha }$ is a colimit diagram in $\operatorname{\mathcal{C}}_{\alpha }$. Then $\operatorname{\mathcal{C}}$ is an $\infty $-category and $\overline{F}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.
Proof. The first assertion follows from Remark 1.4.0.9. To prove the second, set $F = \overline{F}|_{K}$ and $F_{\alpha } = ( \overline{F}_{\alpha } )|_{K}$ for each $\alpha \in A$. By virtue of Proposition 7.1.3.12, it will suffice to show that the restriction map $\theta : \operatorname{\mathcal{C}}_{ \overline{F} / } \rightarrow \operatorname{\mathcal{C}}_{F/}$ is a trivial Kan fibration. Using Lemma 9.1.10.6, we can realize $\theta $ as a filtered colimit of restriction maps $\theta _{\alpha }: (\operatorname{\mathcal{C}}_{\alpha })_{ \overline{F}_{\alpha } / } \rightarrow (\operatorname{\mathcal{C}}_{\alpha })_{ F_{\alpha } / }$. Our hypothesis guarantees that each $\theta _{\alpha }$ is a trivial Kan fibration (Proposition 7.1.3.12), so that $\theta $ is also a trivial Kan fibration (Remark 1.5.5.3). $\square$