Proposition 11.5.0.2. Let $f: K \rightarrow \operatorname{\mathcal{S}}$ be a diagram. Then:
An extension $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}$ is a limit diagram if and only if it is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$.
An extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{S}}$ is a colimit diagram if and only if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$.