Corollary 7.2.2.5. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty $-categories and let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $Y$. Then:
If $Y$ is an initial object of $\operatorname{\mathcal{C}}$, then a diagram $\operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-limit diagram if and only if it carries $\{ Y\} ^{\triangleleft } \simeq \Delta ^1$ to a $U$-cartesian morphism of $\operatorname{\mathcal{D}}$.
If $Y$ is a final object of $\operatorname{\mathcal{K}}$, then a diagram $\operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-colimit diagram if and only if it carries $\{ Y\} ^{\triangleright } \simeq \Delta ^1$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.