$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition Let $X$ be the inverse limit of a tower of simplicial sets

\[ \cdots \xrightarrow {q_3} X(3) \xrightarrow {q_2} X(2) \xrightarrow {q_1} X(1) \xrightarrow {q_0} X(0). \]

If each of the morphisms $q_ n$ is a Kan fibration, then the canonical map $q: X \rightarrow X(0)$ is a Kan fibration.

Proof. This is a special case of Proposition (which guarantees that the collection of all Kan fibrations is closed under transfinite composition in the opposite of the category of simplicial sets). $\square$