Let $R$ be a ring, and let $B_\bullet$, $C_\bullet$, $D_\bullet$, and $E_\bullet$ be chain complexes of left $R$-modules, with differentials denoted respectively by
Let $f,g: C_\bullet \to D_\bullet$ be chain maps, with degree-$n$ components $f_n,g_n: C_n \to D_n$, and suppose that $f$ and $g$ are chain homotopic. Thus there exists a family of $R$-linear maps
If $a: B_\bullet \to C_\bullet$ and $b: D_\bullet \to E_\bullet$ are chain maps, then the composite chain maps $f \circ a$ and $g \circ a$ from $B_\bullet$ to $D_\bullet$ are chain homotopic, and the composite chain maps $b \circ f$ and $b \circ g$ from $C_\bullet$ to $E_\bullet$ are chain homotopic.