Universal Property of Quotient Groups (Hungerford)

If $f:G\to H$ is a homomorphism and $N$ is a normal subgroup of $G$ contained in the kernel of $f$ , then $f$ “factors through” the quotient $G/N$ uniquely.

This can be used to prove the following proposition:
A chain map $f_\bullet$ between chain complexes $(A_\bullet, \partial_{A, \bullet})$ and $(B_\bullet, \partial_{B,\bullet})$ induces homomorphisms between the homology groups of the two complexes.

Proof:
The relation $\partial f=f\partial$ implies that $f$ takes cycles to cycles since $\partial\alpha=0$ implies $\partial(f\alpha)=f(\partial\alpha)=0$ . Also $f$ takes boundaries to boundaries since $f(\partial\beta)=\partial(f\beta)$ . Hence $f_\bullet$ induces a homomorphism $(f_\bullet)_*: H_\bullet (A_\bullet)\to H_\bullet (B_\bullet)$ , by universal property of quotient groups.

For $\beta\in\text{Im} \partial_{A,n+1}$ , we have $\pi_{B,n}f_n(\beta)=\text{Im}\partial_{B,n+1}$ . Therefore $\text{Im}\partial_{A,n+1}\subseteq\ker(\pi_{B,n}\circ f_n)$ .