Let $R$ be a ring, let $C_\bullet = (C_n, d_n^C)_{n \in \mathbb Z}$ and $D_\bullet = (D_n, d_n^D)_{n \in \mathbb Z}$ be chain complexes of left $R$-modules, and let $f, g: C_\bullet \to D_\bullet$ be chain maps. Suppose that $f$ and $g$ are chain homotopic, meaning that there exists a family of $R$-linear maps $h_n: C_n \to D_{n+1}$ for $n \in \mathbb Z$ such that, for every $n \in \mathbb Z$,

\begin{align*} f_n - g_n = d_{n+1}^D \circ h_n + h_{n-1} \circ d_n^C. \end{align*}

Then, for every $n \in \mathbb Z$, the induced homomorphisms on homology are equal:

\begin{align*} H_n(f) = H_n(g): H_n(C_\bullet) \to H_n(D_\bullet). \end{align*}