Lemma 12.13.1. Let $\mathcal{A}$ be an additive category. Let $f, g : B_\bullet \to C_\bullet $ be morphisms of chain complexes. Suppose given morphisms of chain 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 $c \circ g \circ a$.

** Hom functors of $\text{Ch}(\mathcal{A})$ respect the homotopy relation. **

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

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## Comments (1)

Comment #7383 by ElĂas Guisado on

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