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

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 $c \circ g \circ a$.

Proof. Omitted. $\square$

Comment #7383 by Elías Guisado on

Suggested slogan: Hom functors of $\operatorname{Ch}(\mathcal{A})$ respect the homotopy relation.

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