Corollary 4.4.3.21 (Slices of Isofibrations). Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories. Then, for every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the induced map of slice $\infty $-categories $U_{/f}: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{D}}_{ / (U \circ f)}$ is also an isofibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The functor $U_{/f}$ factors as a composition
\[ \operatorname{\mathcal{C}}_{/f} \xrightarrow {T} \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / (U \circ f) } \xrightarrow {U'} \operatorname{\mathcal{D}}_{ / (U \circ f) } \]
where $T$ is a right fibration (Proposition 4.3.6.8) and $U'$ is a pullback of $U$. Example 4.4.1.11 guarantees that $T$ is an isofibration, and Corollary 4.4.3.19 guarantees that $U'$ is an isofibration. Applying Remark 4.4.1.10, we conclude that $U_{/f} = U' \circ T$ is an isofibration. $\square$