Variant 7.6.6.11 (Towers of Kan Fibrations). Suppose we are given a collection of Kan complexes $\{ X(n) \} _{n \geq 0}$ and morphisms $f_ n: X(n+1) \rightarrow X(n)$, which we view as a tower
\[ \cdots \rightarrow X(4) \xrightarrow { f_3 } X(3) \xrightarrow { f_2 } X(2) \xrightarrow { f_1} X(1) \xrightarrow { f_0 } X(0). \]
If each of the morphisms $f_ n$ is a Kan fibration, then the limit $\varprojlim _{n} X(n)$ (formed in the ordinary category of simplicial sets) is also a Kan complex, which can be also be viewed as a limit of the associated tower $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{S}}$ (combine Example 7.6.6.10 with Proposition 7.4.5.1).