Remark 4.5.5.7. Let $q: X \rightarrow S$ be an isofibration of simplicial sets. Then $q$ is an inner fibration: that is, it has the right lifting property with respect to every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^{n}$ for $0 < i < n$ (such inclusions are categorical equivalences, by virtue of Corollary 4.5.3.14). In particular, for each vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is an $\infty $-category (Remark 4.1.1.6). Moreover, if $S$ is an $\infty $-category, then $X$ is also an $\infty $-category (Remark 4.1.1.9).

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