Kerodon

Remark 4.8.1.18. Let $\{ \operatorname{\mathcal{C}}_{j} \} _{j \in \operatorname{\mathcal{J}}}$ be a diagram of simplicial sets having limit $\operatorname{\mathcal{C}}= \varprojlim _{j \in \operatorname{\mathcal{J}}} \operatorname{\mathcal{C}}_ j$ and let $n$ be an integer. If each $\operatorname{\mathcal{C}}_{j}$ is an $(n,1)$-category and $\operatorname{\mathcal{C}}$ is an $\infty $-category, then $\operatorname{\mathcal{C}}$ is also an $(n,1)$-category. For $n \leq -2$, this is trivial (Example 4.8.1.10). The case $n \geq -1$ follows from Remarks 4.8.1.14 and 4.8.1.16, since $\operatorname{\mathcal{C}}$ can be identified with a simplicial subset of the product $\prod _{j \in J} \operatorname{\mathcal{C}}_{j}$.