Corollary 9.4.8.12. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be accessible $\infty $-category and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an accessible functor. If $\operatorname{\mathcal{D}}_0$ is an accessibly embedded subcategory of $\operatorname{\mathcal{D}}$, then the inverse image $\operatorname{\mathcal{C}}_0 = F^{-1}( \operatorname{\mathcal{D}}_0 )$ is an accessibly embedded subcategory of $\operatorname{\mathcal{D}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We have a pullback diagram
9.15
\begin{equation} \begin{gathered}\label{equation:pullback-of-accessibly-embedded} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \ar [r] \ar [d] & \operatorname{\mathcal{D}}_0 \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{D}}, } \end{gathered} \end{equation}
where the right vertical map is an accessible isofibration, so that (9.15) is a categorical pullback square (Corollary 4.5.3.28). Invoking Corollary 9.4.8.11, we deduce that $\operatorname{\mathcal{C}}_0$ is an accessible $\infty $-category and the left vertical map is an accessible isofibration: that is, $\operatorname{\mathcal{C}}_0$ is an accessibly embedded subcategory of $\operatorname{\mathcal{C}}$. $\square$