Corollary 7.1.5.21. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, and let $C \in \operatorname{\mathcal{C}}$ be an object having image $D = U(C)$ in $\operatorname{\mathcal{D}}$. If the object $C$ is $U$-initial, then it is initial when regarded as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. The converse holds if $U$ is a cartesian fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Apply Proposition 7.1.5.19 in the special case where $\operatorname{\mathcal{E}}= \operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}' = \{ D\} $. $\square$