From Rotman's Algebraic Topology:
Assume $X = X_1^{\circ} \cup X_2^{\circ}$. If $D: H_n(X) \rightarrow H_{n-1}(X_1 \cap X_2)$ is the connecting homomorphism in the Mayer-Vietoris sequence, then $D(\text{cls }z) = D(\text{cls }(\gamma_1 + \gamma_2)) = \text{cls } \partial \gamma_1 $.
Proof:
Define $h_{\text{#}} : S_n(X_1) / S_n(X_1 \cap X_2) \rightarrow S_n(X) + S_n(X_2) / S_n(X_2)$ and $q_{\text{#}} : S_n(X) \rightarrow S_n(X) + S_n(X_2) / S_n(X_2)$, where $q_{\text{#}} : \gamma_1 + \gamma_2 \mapsto \gamma_1 + S_n(X_2)$ and $H_{\text{#}}$ is the isomorphism of the second isomorphism theorem. Hence $h_{\text{#}}^{-1}$ sends the coset $\gamma_1 + \gamma_2 + S_n(X)$ to $\gamma_1 + S_n(X_1 \cap X_2)$.
The author then states:
The formula for $D$ for the connecting homomorphism of the Mayer-Vietoris sequence is $D = d{h_{\text{#}}^{-1}} q_{\text{#}}$, where $d$ is the connecting homomorphism from the exact sequence: $0 \rightarrow S_*(X_1 \cap X_2) \rightarrow S_*(X_1) \rightarrow S_*(X_1) / S_*(X_1 \cap X_2) \rightarrow 0$.
This problem with this is that he defined $D$ as the connecting homomorphism of the Mayer Vietoris sequence by $D = d' h_*^{'-1}q_*'$, where $d'$ is the connecting homomorphism of the pair $(X_1, X_1 \cap X_2)$, and $h_*^{'-1} : H_n(X, X_2) \rightarrow H_n(X_1, X_1 \cap X_2)$ and $q_*' : H_n(X) \rightarrow H_n(X, X_2)$.
But $h \neq h'$ and $q \neq q'$ and $d \neq d'$. The rest of the proof makes sense if we just let $D$ be the function defined above but I don't see it's relationship to the Mayer-Vietoris sequence. $q$ and $h$ have the same domains as $q'$ and $h'$, respectively, but their comdomains are not equal: $S_n(X_1) + S_n(X_2) / S_n(X_2) \neq S_n(X) / S_n(X_2)$.
Is there a relationship between $S_n(X_1) + S_n(X_2) / S_n(X_2) $ and $ S_n(X) / S_n(X_2)$ that I'm missing or does this theorem have nothing to do with the Mayer-Vietoris sequence?