Proposition 2.5.1.8. Let $C_{\ast }$ and $D_{\ast }$ be chain complexes with values in an abelian category $\operatorname{\mathcal{A}}$, and let $f,f': C_{\ast } \rightarrow D_{\ast }$ be a pair of chain maps. If $f$ and $f'$ are chain homotopic, then they induce the same map from $\mathrm{H}_{n}(C)$ to $\mathrm{H}_{n}(D)$ for every integer $n$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $h = \{ h_ m \} _{m \in \operatorname{\mathbf{Z}}}$ be a chain homotopy from $f$ to $f'$, so that $f'_{n} - f_ n = \partial _{D} \circ h_{n} + h_{n-1} \circ \partial _{C}$. It follows that, when restricted to the subobject $\mathrm{Z}_{n}(C) \subseteq C_ n$, the difference $f'_{n} -f_ n = \partial _{D} \circ h_ n$ factors through the subobject $\mathrm{B}_{n}(D) \subseteq \mathrm{Z}_{n}(D)$, so the induced maps $\mathrm{H}_{n}(f), \mathrm{H}_{n}(f'): \mathrm{H}_{n}(C) \rightarrow \mathrm{H}_{n}(D)$ are the same. $\square$