The left and right projective constructions of Tor are naturally isomorphic

The augmentations give degreewise surjective chain maps $a:\mathrm{Tot}K\to N{\otimes }_R{P}_{\bullet }$ and $b:\mathrm{Tot}K\to Q_{\bullet }{\otimes }_{R}M$, supported respectively on $p=0$ and $q=0$. Their surjectivity and exact augmented fixed-$q$ and fixed-$p$ complexes follow from The augmented fixed-q rows of the tensor double complex are exact and The augmented rows of the tensor double complex are exact, respectively. The differential is $d_h+d_v$, with $d_v=(-1{)}^p(1\otimes d_P)$ as in The first-quadrant tensor double complex of two projective resolutions.

The kernel of $a$ is the total of the double complex obtained by replacing $K_{0,q}$ by $\ker (Q_0\otimes P_q\to N\otimes P_q)$. Every fixed-$q$ horizontal complex is exact. For a total cycle $z$ of degree $n\ge 0$, take its largest nonzero $q$ component $z_{p,q}$. The cycle equation at that $q$ says $d_h{z}_{p,q}=0$, because the next higher $q$ component is zero. Horizontal exactness gives $y_{p+1,q}$ with $d_{h}y=z_{p,q}$. Subtracting $dy$ removes that component and introduces only a component at $q-1$. Repeating terminates at $q=0$ and expresses $z$ as a boundary. Thus $\ker a$ is acyclic, including degree zero; negative degrees are zero. This is the filtration by $q$, not by $p$.

The kernel of $b$ is obtained by replacing $K_{p,0}$ by $\ker (Q_p\otimes P_0\to Q_p\otimes M)$. Its fixed-$p$ vertical complexes are exact, with multiplication of their differentials by $(-1{)}^p$ harmless. Now eliminate a cycle's largest $p$ component by solving $d_{v}y=z_{p,q}$ in bidegree $(p,q+1)$. Subtracting $dy$ leaves only lower $p$ components; finitely many repetitions show that $\ker b$ is acyclic. This uses the filtration by $p$, not by $q$.

The short exact sequences of each kernel, total complex and edge, together with The long exact sequence in homology, make both $a$ and $b$ quasi-isomorphisms. Hence $H_i(N{\otimes }_R{P}_{\bullet })\stackrel{H_i(a)}{\leftarrow }{H}_i(\mathrm{Tot}K)\stackrel{H_i(b)}{\to }{H}_i(Q_{\bullet }{\otimes }_{R}M)$ gives the asserted isomorphism $H_i(b)H_i(a{)}^{-1}$.

Under DC, maps of modules lift to maps of their supplied resolutions by Projective comparison maps exist. The resulting tensor double-complex maps commute with $a,b$, proving naturality of the ratio in step 3.1. Two lifts are chain-homotopic by Projective comparison maps are unique up to chain homotopy. Tensor homotopies in the first factor are $h_Q\otimes 1$ and in the second factor $(-1{)}^{p}1\otimes h_P$; direct substitution gives the total homotopy equation. Thus induced homology maps are independent of lifts. Taking module identities also gives independence and coherence under change of the supplied resolutions.