Proposition 9.2.9.14. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{E}}$ be an $\infty $-category equipped with a $(\kappa ,\lambda )$-cocomplete inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$. Let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{E}}$, where $U(X) = 0$ and $U(Y) = 1$. If $X$ and $Y$ are $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{E}}$, then $f$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{Fun}_{ / \Delta ^1 }( \Delta ^1, \operatorname{\mathcal{E}})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The $\infty $-category $\operatorname{Fun}_{ / \Delta ^1}( \Delta ^1, \operatorname{\mathcal{E}})$ can be identified with the oriented fiber product $\operatorname{\mathcal{E}}_0 \vec{\times }_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_1$. The desired result now follows from Proposition 9.2.8.3. $\square$