Here is the link to the book: http://pi.math.cornell.edu/~hatcher/AT/AT.pdf
In the proof of Theorem 4.37(p.372), there is a huge diagram and the picture below is a portion of it:
The definition of the groups $\pi_n'$ are explained in the last paragraph in p.370.
I can't see where the map $\partial '$ came from. It seems that it is induced by the map $\partial$. However, in order to $\partial$ passes to the quotient and induce $\partial'$, the image under $\partial$ of the kernel of the map $\pi_{n+1}(X,X^n \cup A) \to \pi_{n+1}'(X,X^n \cup A)$ must be contained in the kernel of the map $\pi_n(X^n\cup A,A)\to \pi_n'(X^n \cup A,A)$. I tried to verify this, but I found that this is highly nontrivial. Am I missing something?
Thanks in advance.