Kerodon

Proposition 4.7.3.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then $X$ is an initial object of $\operatorname{\mathcal{C}}$ if and only if $F(X)$ is an initial object of $\operatorname{\mathcal{D}}$. Similarly, $X$ is a final object of $\operatorname{\mathcal{C}}$ if and only if $F(X)$ is a final object of $\operatorname{\mathcal{D}}$.