Let $R$ be a ring, and let $C_\bullet = (C_n, d^C_n)_{n \in \mathbb{Z}}$ and $D_\bullet = (D_n, d^D_n)_{n \in \mathbb{Z}}$ be chain complexes of left $R$-modules. Let $\operatorname{Ch}_R(C_\bullet, D_\bullet)$ denote the set of chain maps $f_\bullet: C_\bullet \to D_\bullet$.
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For $f_\bullet, g_\bullet \in \operatorname{Ch}_R(C_\bullet, D_\bullet)$, define $f_\bullet \sim g_\bullet$ if there exists a family of $R$-linear maps