Example 7.1.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $Y \in \operatorname{\mathcal{C}}$ is final (in the sense of Definition 4.6.7.1) if and only if the map
\[ (\emptyset )^{\triangleleft } \simeq \Delta ^0 \xrightarrow {Y} \operatorname{\mathcal{C}} \]
is a limit diagram in $\operatorname{\mathcal{C}}$. Similarly, $Y$ is initial if and only if the map
\[ (\emptyset )^{\triangleright } \simeq \Delta ^0 \xrightarrow {Y} \operatorname{\mathcal{C}} \]
is a colimit diagram in $\operatorname{\mathcal{C}}$.