$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 3.5.8.9. Let $X$ be a Kan complex. Then the transition maps in the canonical Postnikov tower
\[ \cdots \rightarrow \pi _{\leq 3}(X) \rightarrow \pi _{\leq 2}(X) \rightarrow \pi _{\leq 1}(X) \rightarrow \pi _{\leq 0}(X) \]
are Kan fibrations. Moreover, for every vertex $x \in X$ and every integer $n > 0$, there is a canonical homotopy equivalence
\[ \mathrm{K}(G,n) \rightarrow \{ x\} \times _{ \pi _{\leq n-1}(X) } \pi _{\leq n}(X), \]
for $G = \pi _{n}(X,x)$.
Proof.
For $n > 0$, the transition map $\pi _{\leq n}(X) \rightarrow \pi _{\leq n-1}(X)$ factors as a composition
\[ \pi _{\leq n}(X) \xrightarrow {f_ n} \operatorname{cosk}_{n-1}^{\circ }(X) \xrightarrow {g} \pi _{\leq n-1}(X), \]
where $f_{n}$ is the Kan fibration of Proposition 3.5.8.7 and $g$ is the trivial Kan fibration of Corollary 3.5.6.14. We therefore obtain a pullback diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \mathrm{K}(G,n) \ar [r] \ar [d] & \{ x\} \times _{ \pi _{\leq n-1}(X) } \pi _{\leq n}(X) \ar [d] \\ \{ x\} \ar [r] & \{ x\} \times _{ \pi _{\leq n-1}(X) } \operatorname{cosk}_{n-1}^{\circ }(X) } \]
where the vertical maps are Kan fibrations and the lower right corner is contractible. In particular, the lower horizontal map is a homotopy equivalence. Applying Corollary 3.4.1.5, we deduce that the upper horizontal map is also a homotopy equivalence.
$\square$