Dold's proof of equivalence singular and cellular homology

Question

I would like to ask for some help understanding a claim in Dold's proof of the equivalence of cellular and singular homology. The point is that I don't get why $\delta_n=j_*\delta_*$ where: $\delta_n:H_n (X^n, X^{n-1}) \to H_{n-1} (X^{n-1}, X^{n-2}) $, $j_*:H_{n-1} (X^{n-1}, X^k)\to H_{n-1} (X^{n-1}, X^{n-2})$ and $\delta_*:H_n (X^n, X^{n-1})\to H_{n-1} (X^{n-1}, X^k)$. Also assume that $ X_k\subseteq X_{n-2}\subseteq X_{n-1}\subseteq X_{n}$. Dold says that this is consequence of naturality of $\delta_*$, the connecting homomorphism of the long exact sequence of the triple, but I just can't see any morphism of short exact sequences that give the desired result. In case more context is needed, I'm talking about the proof of proposition 1.3 in page 87. Thanks in advance for any hint you could provide.