Chain map

Remarks

Chain maps are composed degreewise, and the identity chain map has $f_n = 1_{C_n}$. With these, chain complexes in $\mathcal{A}$ form a category, often written $\mathrm{Ch}(\mathcal{A})$. A chain map sends $\ker d^C_n$ into $\ker d^D_n$ and $\im d^C_{n+1}$ into $\im d^D_{n+1}$.

Example: let $C_\bullet = D_\bullet$ be the complex with $C_1 = C_0 = \Z$, $d_1$ multiplication by $3$, and $C_n = 0$ otherwise. Then $f_1 = f_0 =$ multiplication by $2$ is a chain map, since $3 \cdot 2x = 2 \cdot 3x$ for $x \in \Z$.

Non-example: in the same complex, take $f_1 = 1_{\Z}$ and $f_0 = 0$. Then $d_1 \circ f_1$ is multiplication by $3$ but $f_0 \circ d_1 = 0$, so the square in degree $1$ does not commute.