Remark 2.5.0.7. Let $C_{\ast }$ and $D_{\ast }$ be chain complexes with values in an additive category $\operatorname{\mathcal{A}}$. Then chain homotopy determines an equivalence relation on the set of chain maps $f: C_{\ast } \rightarrow D_{\ast }$. More precisely:
Every chain map $f: C_{\ast } \rightarrow D_{\ast }$ is chain homotopic to itself, via the chain homotopy given by the collection of zero maps $\{ 0: C_{n} \rightarrow D_{n+1} \} $.
Let $f,f': C_{\ast } \rightarrow D_{\ast }$ be chain maps. If $f$ is chain homotopic to $f'$, then $f'$ is chain homotopic to $f$. More precisely, if $h$ is a chain homotopy from $f$ to $f'$, then $-h$ is a chain homotopy from $f'$ to $f$.
Let $f,f',f'': C_{\ast } \rightarrow D_{\ast }$ be chain maps. If $f$ is chain homotopic to $f'$ and $f'$ is chain homotopic to $f''$, then $f$ is chain homotopic to $f''$. More precisely, if $h$ is a chain homotopy from $f$ to $f'$ and $h'$ is a chain homotopy from $f'$ to $f''$, then $h + h'$ is a chain homotopy from $f$ to $f''$.