Remark 2.5.0.8. Let $C_{\ast }$ and $D_{\ast }$ be chain complexes with values in an additive category $\operatorname{\mathcal{A}}$, and let $f,f': C_{\ast } \rightarrow D_{\ast }$ be chain maps which are chain homotopic. Then:
For every chain map $g: D_{\ast } \rightarrow E_{\ast }$, the composite maps $g \circ f$ and $g \circ f'$ are chain homotopic. More precisely, if $h = \{ h_ n \} _{n \in \operatorname{\mathbf{Z}}}$ is a chain homotopy from $f$ to $f'$, then the collection of composite maps $\{ g_{n+1} \circ h_{n} \} $ is a chain homotopy from $g \circ f$ to $g \circ f'$.
For every chain map $e: B_{\ast } \rightarrow C_{\ast }$, the composite maps $f \circ e$ and $f' \circ e$ are chain homotopic. More precisely, if $h = \{ h_ n \} _{n \in \operatorname{\mathbf{Z}}}$ is a chain homotopy from $f$ to $f'$, then the collection of composite maps $\{ h_ n \circ e_ n \} $ is a chain homotopy from $f \circ e$ to $f' \circ e$.