Kerodon

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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 an initial 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 $U$-limit diagram.