Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 7.1.6.8. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{U} \ar [r]^-{F} & \operatorname{\mathcal{C}}' \ar [d]^{U'} \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{D}}', } \]

where the horizontal maps are equivalences of $\infty $-categories. Then an object $X \in \operatorname{\mathcal{C}}$ is $U$-initial if and only if $F(X) \in \operatorname{\mathcal{C}}'$ is $U'$-initial, and $U$-final if and only if $F(X)$ is $U'$-final.