The Tor balance isomorphism is natural and coherent under a change of resolutions

Form $K_{p,q}=Q_p{\otimes }_R{P}_q$ as in The first-quadrant tensor double complex of two projective resolutions. The augmentations define degreewise surjective chain maps $a:\mathrm{Tot}K\to N{\otimes }_{R}P$ and $b:\mathrm{Tot}K\to Q{\otimes }_{R}M$.

Choose comparison maps $g:Q\to Q^{\mathrm{\prime }}$ and $f:P\to P^{\mathrm{\prime }}$ lifting $u,v$ by Projective comparison maps exist. The map $T=\mathrm{Tot}(g\otimes f)$ commutes with both augmented edge maps: $a^{\mathrm{\prime }}T=(u\otimes f)a$ and $b^{\mathrm{\prime }}T=(g\otimes v)b$. These identities hold on pure tensors because comparison maps preserve augmentations.

By The augmented fixed-q rows of the tensor double complex are exact, each fixed-$q$ augmented horizontal complex is exact. If $z\in \ker a$ is a total cycle, choose its largest nonzero $q$-component. The cycle equation there is a horizontal cycle equation, so horizontal exactness gives a preimage one horizontal degree higher. Subtracting its total boundary removes that component and introduces terms only at lower $q$. Finite diagonal support makes the iteration terminate, so $\ker a$ is acyclic. The long exact homology sequence The long exact sequence in homology therefore makes $H(a)$ an isomorphism.

By The augmented rows of the tensor double complex are exact, the same elimination with the largest nonzero $p$-component proves that $\ker b$ is acyclic; the sign $(-1{)}^p$ on the vertical differential does not affect exactness. Hence $H(b)$ is also an isomorphism. Thus the balance map is the specified ratio $H(b)H(a{)}^{-1}$, agreeing with the natural isomorphism asserted in The left and right projective constructions of Tor are naturally isomorphic.

Taking homology in step 1.2 and using steps 2.1--2.2 gives $H(g\otimes v)H(b)H(a{)}^{-1}=H(b^{\mathrm{\prime }})H(a^{\mathrm{\prime }}{)}^{-1}H(u\otimes f)$. This is the naturality square.

Under DC, Projective comparison maps are unique up to chain homotopy makes any two lifts homotopic, and Each resolution-defined Tor construction is covariant in both variables makes their edge homology maps equal. Apply step 3.1 to identity module maps for replacement resolutions. Opposite comparison maps compose to a lift of the identity, so they induce inverse identifications; a direct and a composite comparison among three resolutions lift the same identity and therefore induce the same map. Identity comparisons induce identities. This proves independence and the cocycle laws.