In Allen Hatcher's book on page 117 (bottom) he says the following:
The boundary map $\partial : H_n (X, A)\rightarrow H_{nā1} (A)$ has a very simple description: If a class $[\alpha] \in H_n(X,A)$ is represented by a relative cycle $\alpha$, then $\partial[\alpha]$ is the class of the cycle $\partial \alpha$ in $H_{nā1} (A)$ . This is immediate from the algebraic definition of the boundary homomorphism in the long exact sequence of homology groups associated to a short exact sequence of chain complexes.
I can see that $\partial \alpha \in C_{n-1}(A)$ and that it must be a cycle since $\partial\partial = 0$. But how is the relation $\partial[\alpha] = [\partial \alpha]$ established (Left hand side contains the $\partial : H_n (X, A)\rightarrow H_{nā1} $, Right hand side contains $\partial : C_n(A) \rightarrow C_{n-1}(A)$).