Kerodon

Remark 7.2.4.6. Let $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ be a filtered diagram of simplicial sets, where each $\operatorname{\mathcal{C}}_{\alpha }$ is a filtered $\infty $-category. Then the colimit $\operatorname{\mathcal{C}}= \varinjlim _{\alpha } \operatorname{\mathcal{C}}_{\alpha }$ is also a 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 finite 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 $ (see Proposition 3.6.1.9). Our assumption that $\operatorname{\mathcal{C}}_{\alpha }$ is filtered 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 $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.