Corollary 7.2.1.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $X$ be an initial object of $\operatorname{\mathcal{C}}$. Then $F$ is left cofinal if and only if $F(X)$ is an initial object of $\operatorname{\mathcal{D}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$