Kerodon

Remark 9.1.6.11. Let $\kappa $ be an infinite cardinal and let $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ be a $\kappa $-filtered diagram of simplicial sets, where each $\operatorname{\mathcal{C}}_{\alpha }$ is a $\kappa $-filtered $\infty $-category. Then the colimit $\operatorname{\mathcal{C}}= \varinjlim _{\alpha } \operatorname{\mathcal{C}}_{\alpha }$ (formed in the category of simplicial sets) is also a $\kappa $-filtered $\infty $-category. To prove this, we first observe that $\operatorname{\mathcal{C}}$ is an $\infty $-category (Remark 1.4.0.9). If $K$ is a $\kappa $-small simplicial set, then any morphism $f: K \rightarrow \operatorname{\mathcal{C}}$ factors through $f_{\alpha }: K \rightarrow \operatorname{\mathcal{C}}_{\alpha }$ for some index $\alpha $ (by virtue of our assumption that the index diagram is $\kappa $-filtered). Our assumption that $\operatorname{\mathcal{C}}_{\alpha }$ is $\kappa $-filtered then guarantees that $f_{\alpha }$ extends to a diagram $\overline{f}_{\alpha }: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\alpha }$, from which it follows that $f$ extends to a diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.