Corollary 4.8.6.34 (Transitivity). Let $n$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be inner fibrations of simplicial sets, where $G$ is $n$-categorical. Then $F$ is $n$-categorical if and only if $G \circ F$ is $n$-categorical.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For $n < 0$, this follows immediately from the definitions. We may therefore assume that $n \geq 0$. Using Remark 4.8.6.33, we can reduce to the case where $\operatorname{\mathcal{E}}= \Delta ^ m$ is a standard simplex. In this case, our assumption on $G$ guarantees that $\operatorname{\mathcal{D}}$ is an $(n,1)$-category. We wish to show that $\operatorname{\mathcal{C}}$ is an $(n,1)$-category if and only if the inner fibration $F$ is $n$-categorical, which follows from Proposition 4.8.6.32. $\square$