Tor admits dimension shifting in either variable

Assume Dependent Choice (DC) and supply the projective resolutions used to compute Tor. If $0\to K\to P\to M\to 0$ is exact in left $R$-modules with $P$ projective, then for every right module $N$ and $i\ge 1$ the connecting map gives ${\mathrm{Tor}}_{i+1}^R(N,M)\cong {\mathrm{Tor}}_i^R(N,K)\mathrm{.}$ Similarly, for $0\to L\to Q\to N\to 0$ exact in right modules with $Q$ projective and a left module $M$, the connecting map gives ${\mathrm{Tor}}_{i+1}^R(N,M)\cong {\mathrm{Tor}}_i^R(L,M)(i\ge 1)\mathrm{.}$ These isomorphisms are natural for maps of the respective short exact sequences and of the other module; they do not assert the same formula at $i=0$.

The length-zero computation and the inference that an exact segment $0\to A\to B\to 0$ gives an isomorphism are choice-free. DC is used here for the quoted natural Tor sequences and their resolution comparisons. If those sequences and coherent identifications are supplied explicitly, the same dimension-shifting inference needs no additional choice axiom.