# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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 $\operatorname{\mathcal{D}}$.

Proof. Apply Corollary 7.2.2.5 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$ (and use Example 5.1.1.4). $\square$