If $(X_\alpha,A_\alpha)$ are disjoint topological pairs, is the following statement true? $$ \bigoplus_\alpha H_n(X_\alpha,A_\alpha)=H_n\left(\bigcup_\alpha X_\alpha,\bigcup_\alpha A_\alpha\right) $$
I would argue that this is true because \begin{align} \bigoplus_\alpha H_n(X_\alpha,A_\alpha) &= \bigoplus_\alpha \tilde H_n(X_\alpha/A_\alpha) = \tilde H_n\left(\bigvee_\alpha(X_\alpha/A_\alpha)\right)\\ &= \tilde H_n\left(\bigcup_\alpha X_\alpha/\bigcup_\alpha A_\alpha\right) = H_n\left(\bigcup_\alpha x_\alpha,\bigcup_\alpha A_\alpha\right).\end{align}
Is this argument correct? Do I have to assume here that $A_\alpha\subset X_\alpha$ is a cofibration?
Another approach of which I thought was to look at the commutative (?) diagram $$\require{AMScd}\begin{CD} \oplus_\alpha H_n(A_\alpha) @>i_*>> \oplus_\alpha H_n(X_\alpha) @>j_*>> \oplus_\alpha H_n(X_\alpha,A_\alpha) @>\partial_*>>\oplus_\alpha H_{n-1}(A_\alpha) \dots \\ @V\cong VV @V\cong VV @VVV @V\cong VV \\ H_n(\cup_\alpha A_i) @>i_*>>H_n(\cup_\alpha X_\alpha) @>j_*>> H_n(\cup_\alpha X_\alpha, \cup_\alpha A_\alpha) @>\partial_*>> H_{n-1}(\cup_\alpha A_\alpha)\dots \end{CD}$$ and argue by the five lemma that the third vertical homomorphism is indeed an isomorphism. Maybe I am overcomplicating things, but I did not find simpler arguments.