Warning 5.6.3.6. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets. In the language of ยง4.8, Corollary 5.6.3.5 asserts that if each of the $\infty $-categories $\mathscr {F}(C)$ is a $(1,1)$-category, then the cocartesian fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is $1$-categorical (see Example 4.8.6.27). Beware that for $n \geq 2$, the assumption that each $\mathscr {F}(C)$ is an $(n,1)$-category does not guarantee that the cocartesian fibration $U$ is $n$-categorical. However, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then it is essentially $n$-categorical: this is an immediate consequence of Variant 5.1.5.17.
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