Corollary 7.2.3.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories. Then:
If $\operatorname{\mathcal{C}}$ has an initial object $Y$ and $\overline{F}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a functor which carries $\{ Y\} ^{\triangleleft } \simeq \Delta ^1$ to an isomorphism in $\operatorname{\mathcal{D}}$, then $\overline{F}$ is a $U$-limit diagram.
If $\operatorname{\mathcal{C}}$ has a final object $Y$ and $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a functor which carries $\{ Y\} ^{\triangleright } \simeq \Delta ^1$ to an isomorphism in $\operatorname{\mathcal{D}}$, then $\overline{F}$ is a $U$-colimit diagram.