Corollary 3.2.4.15. Let $X$ be the colimit of a diagram of Kan complexes
\[ X(0) \xrightarrow { f_0 } X(1) \xrightarrow { f_1 } X(2) \xrightarrow { f_2 } \cdots \]
If each of the morphisms $f_ n$ is nullhomotopic, then $X$ is contractible.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Theorem 3.2.4.3 (and Remark 3.2.4.13), it will suffice to show that every morphism of simplicial sets $e: \operatorname{\partial \Delta }^ m \rightarrow X$ is nullhomotopic. Since $\operatorname{\partial \Delta }^ m$ is a finite simplicial set, the morphism $e$ factors through $X(n)$ for $n \gg 0$. The desired result now follows from our assumption that $f_ n: X(n) \rightarrow X(n+1)$ is nullhomotopic. $\square$