Lemma 12.13.10. Let $\mathcal{A}$ be an additive category. Let

\[ 0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 \]

be a complex (!) of complexes. Suppose that we are given splittings $B^ n = A^ n \oplus C^ n$ compatible with the maps in the displayed sequence. Let $s^ n : C^ n \to B^ n$ and $\pi ^ n : B^ n \to A^ n$ be the corresponding maps. Then the family of morphisms

\[ \pi ^{n + 1} \circ d_ B^ n \circ s^ n : C^ n \to A^{n + 1} \]

define a morphism of complexes $\delta : C^\bullet \to A[1]^\bullet $.

**Proof.**
Denote $i : A^\bullet \to B^\bullet $ and $q : B^\bullet \to C^\bullet $ the maps of complexes in the short exact sequence. Then $i^{n + 1} \circ \pi ^{n + 1} \circ d_ B^ n \circ s^ n = d_ B^ n \circ s^ n - s^{n + 1} \circ d_ C^ n$. Hence $i^{n + 2} \circ d_ A^{n + 1} \circ \pi ^{n + 1} \circ d_ B^ n \circ s^ n = d_ B^{n + 1} \circ (d_ B^ n \circ s^ n - s^{n + 1} \circ d_ C^ n) = - d_ B^{n + 1} \circ s^{n + 1} \circ d_ C^ n$ as desired.
$\square$

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