Kerodon

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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 $\{ X\} ^{\triangleleft } \simeq \Delta ^1$ 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 $\{ X\} ^{\triangleright } \simeq \Delta ^1$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.

Proof. If $Y$ is an initial object of $\operatorname{\mathcal{K}}$, then the inclusion map $\{ Y\} \hookrightarrow \operatorname{\mathcal{K}}$ is left cofinal (Corollary 4.6.6.25). The first assertion now follows by combining Corollary 7.2.2.2 with Example 7.1.5.9. The second assertion follows by a similar argument. $\square$