Corollary 7.2.2.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Then:
If $\operatorname{\mathcal{C}}$ has an initial object $Y$, then a functor $\operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram if and only if it carries $\{ Y\} ^{\triangleleft } \simeq \Delta ^1$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.
If $\operatorname{\mathcal{C}}$ has a final object $Y$, then a functor $\operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram if and only if it carries $\{ Y\} ^{\triangleright } \simeq \Delta ^1$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.