Corollary 11.9.3.5. Let $X$ be the inverse limit of a tower of simplicial sets
\[ \cdots \xrightarrow {q_3} X(3) \xrightarrow {q_2} X(2) \xrightarrow {q_1} X(1) \xrightarrow {q_0} X(0). \]
Assume that each of the simplicial sets $X(n)$ is a contractible Kan complex and that each of the morphisms $q_ n$ is a Kan fibration. Then the inverse limit $X = \varprojlim _{n \geq 0} X(n)$ is a contractible Kan complex.