Corollary 4.4.3.21. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative isofibration of $\infty $-categories. Then, for each object $D \in \operatorname{\mathcal{D}}$, the fiber $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is a Kan complex.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Since $q$ is an inner fibration, the simplicial set $\operatorname{\mathcal{C}}_{D}$ is an $\infty $-category (Remark 4.1.1.6). It will therefore suffice to show that every morphism $f$ in $\operatorname{\mathcal{C}}_{D}$ is an isomorphism (Proposition 4.4.2.1). By virtue of Corollary 4.4.3.20, this is equivalent to the requirement that $f$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$. This follows from our assumption that $q$ is conservative, since $q(f) = \operatorname{id}_{D}$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$. $\square$