Tor, deleted term,exactness, $Tor_0$,exact sequences

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Say we have a setting like this on page 28. I have a few questions on the deletion of $M\otimes N$. One is that we loose by this isomorphism Tor$_0^R(M,N)\cong M\otimes_R N$. The second is why the deleted tensored sequence is exact at the last arrow $\to P_0\otimes N\to 0$? Finally, why did we at all erase the last term $M\otimes_R N$ in front of $0$?