The long exact Tor sequence in the left-module variable

By the DC-qualified The horseshoe lemma for projective resolutions, choose a projective horseshoe resolution of the middle module that fits with resolutions of the outer modules into a degreewise split short exact sequence. DC supplies the compatible choices through its successive degrees. Tensoring with $N$ preserves each splitting: tensor the inclusion, projection and their two section/retraction maps, whose splitting identities persist by additivity and composition. Hence the tensored sequence of complexes is short exact over an arbitrary ring.

Apply The long exact sequence in homology to the complexes in step 1.1. Their homology groups give the Tor presentations of The balanced Tor bifunctor. The canonical degree-zero identifications of Degree-zero Tor is the tensor product in either construction give the stated tensor tail.

For a morphism of short exact sequences, choose comparison maps on the end resolutions under DC as in Each resolution-defined Tor construction is covariant in both variables. The compatible-middle-map clause of The left derived connecting map is independent of the horseshoe resolution and lifts extends them to a commuting horseshoe ladder; that clause uses projectivity and DC, not a property of a particular derived functor. Tensor the ladder with $N$ and apply Naturality of the homology connecting morphism. Applying the same argument to identity module maps compares different horseshoes. The covariance proposition makes homology maps independent of comparison lifts, so the resulting Tor connecting maps are independent and natural. A map of right modules induces a tensor ladder directly and the same homology naturality applies. This proves the asserted natural exact sequence without assuming left exactness of tensoring.