Corollary 7.1.3.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $K$ be an arbitrary simplicial set. Then:
- $(1)$
If $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, then the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ admits $K$-indexed limits. Moreover, a morphism $K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{f/}$ is a limit diagram if and only if its image in $\operatorname{\mathcal{C}}$ is a limit diagram.
- $(2)$
If $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits, then the slice $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ admits $K$-indexed colimits. Moreover, a morphism $K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a colimit diagram if and only if its image in $\operatorname{\mathcal{C}}$ is a colimit diagram.