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Remark Suppose we are given an inverse system of simplicial sets

\[ \cdots \rightarrow X(4) \rightarrow X(3) \rightarrow X(2) \rightarrow X(1) \rightarrow X(0), \]

where each of the transition maps $X(n) \rightarrow X(n-1)$ is an inner fibration. Then each of the projection maps $\varprojlim _{n} X(n) \rightarrow X(m)$ is an inner fibration. In particular, if any of the simplicial sets $X(m)$ is an $\infty $-category, then the inverse limit $\varprojlim _{n} X(n)$ is also an $\infty $-category.