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Under the Dold–Kan correspondence, a $1$-simplex of $K(C_{\ast })$ is a pair $(a,x)\in C_0\oplus C_1$ withThe horn ${\Lambda }_1^2$ consists of the edges $01$ and $12$, joined at vertex $1$. A map from it is therefore a pair of composable $1$-simplices, equivalently a triplewhere the first edge is $(a,x)$ and the second is $(a+{\partial }_{1}x,y)$. Thus

The Dold–Kan correspondence says thatis an equivalence from simplicial abelian groups to nonnegatively graded chain complexes of abelian groups. The restriction of the right adjoint $K$ to ${\mathbf{Ch}}_{\ge 0}(\mathbf{Ab})$ is a quasi-inverse: both the unit $C_{\ast }\to N_{\ast }K(C_{\ast })$ and counit $K(N_{\ast }A)\to A$ are natural isomorphisms.

Regard the given short exact sequence as a degreewise short exact sequence of chain complexes concentrated in degree $n$. The inverse functor in the Dold–Kan correspondence is exact, so it produces a degreewise short exact sequence of simplicial abelian groupsThe last map is degreewise surjective and hence a Kan fibration. Its strict fiber is $K(A_1,n)$, and a strict fiber of a fibration computes the homotopy fiber. This proves the asserted homotopy fiber sequence of pointed Kan complexes.