Lemma 12.14.3. Let $\mathcal{A}$ be an additive category. Suppose that $A_\bullet $ and $B_\bullet $ are chain complexes. Given any morphism of chain complexes $a : A_\bullet \to B_\bullet $ there is a bijection between the set of homotopies from $a$ to $a$ and $\mathop{\mathrm{Mor}}\nolimits _{\text{Ch}(\mathcal{A})}(A_\bullet , B[1]_\bullet )$. More generally, the set of homotopies between $a$ and $b$ is either empty or a principal homogeneous space under the group $\mathop{\mathrm{Mor}}\nolimits _{\text{Ch}(\mathcal{A})}(A_\bullet , B[1]_\bullet )$.

**Proof.**
See above.
$\square$

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