Lemma 12.12.7. Let $\mathcal{A}$ be an additive category. Let $f, g : B^\bullet \to C^\bullet $ be morphisms of cochain complexes. Suppose given morphisms of cochain complexes $a : A^\bullet \to B^\bullet $, and $c : C^\bullet \to D^\bullet $. If $\{ h^ i : B^ i \to C^{i - 1}\} $ defines a homotopy between $f$ and $g$, then $\{ c^{i - 1} \circ h^ i \circ a^ i\} $ defines a homotopy between $c \circ f \circ a$ and $c \circ g \circ a$.

**Proof.**
Omitted.
$\square$

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