I have been reading MacLane's Book on Homology and I have a doubt in the proof of the Kunneth Formula
In $10.5)$ he he says that $\bigoplus D_{m+1}\otimes H_q(L) \cong H_{n+1}(D \otimes L)$, and i cant quite see this is true, we know that since evey $D_{m}$ is flat we have that $D_m \otimes H_q(L) \cong H_q(D_m \otimes L)$ but i dont see how this gives us the rest, i guess my question is why is $\bigoplus H_q(D_{m+1} \otimes L) \cong H_{n+1}(D \otimes L)$?Is this true because of the way we defined the tensor product of complexes, and because the direct sum commutes with homology?
Also i am a little bit confused on how $10.6)$ proves that $\beta$ is natural, so any advice would be helpfull 
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Thanks in advance.