Lemma 12.14.12. Notation and assumptions as in Lemma 12.14.10. Let $\alpha : A^\bullet \to B^\bullet $, $\beta : B^\bullet \to C^\bullet $ be the given morphisms of complexes. Suppose $(s')^ n : C^ n \to B^ n$ and $(\pi ')^ n : B^ n \to A^ n$ is a second choice of splittings. Write $(s')^ n = s^ n + \alpha ^ n \circ h^ n$ and $(\pi ')^ n = \pi ^ n + g^ n \circ \beta ^ n$ for some unique morphisms $h^ n : C^ n \to A^ n$ and $g^ n : C^ n \to A^ n$. Then

$g^ n = - h^ n$, and

the family of maps $\{ g^ n : C^ n \to A[1]^{n - 1}\} $ is a homotopy between $\delta , \delta ' : C^\bullet \to A[1]^\bullet $, more precisely $(\delta ')^ n = \delta ^ n + g^{n + 1} \circ d_ C^ n + d_{A[1]}^{n - 1} \circ g^ n$.

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