Definition 29.6:

Given two chain complexes $G$ and $K$ with boundary maps $\partial ^{\prime}:G_n\longrightarrow G_{n-1}$ and $\partial ^{\prime\prime}:K_n\longrightarrow K_{n-1}$, a chain map $\phi:G\longrightarrow K$ is a sequence of group homomorphisms $\phi_n:G_n\longrightarrow K_n$ ( $n = 0, 1, 2,\dots$) such that

$$\partial _n^{\prime\prime} \circ \phi_n = \phi_{n-1}\circ \partial _n^{\prime} \eqno(29.13)$$

Equation (29.13) may be summarized by declaring that the following diagram commutes:

$\begin{CD} G_n @> \partial _n^{\prime} >> G_{n-1} \\ @V{\phi_n}VV @VV{\phi_{n-1}} V\\ K_n @> \partial _n^{\prime\prime} >> K_{n-1} \\ \end{CD}$(29.14)