Corollary 4.6.7.21. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are equivalent. Then:
The $\infty $-category $\operatorname{\mathcal{C}}$ has an initial object if and only if the $\infty $-category $\operatorname{\mathcal{D}}$ has an initial object.
The $\infty $-category $\operatorname{\mathcal{C}}$ has a final object if and only if the $\infty $-category $\operatorname{\mathcal{D}}$ has a final object.