Factor reversal gives the commutativity chain homotopy

The coefficients of $db\otimes a$ and $b\otimes da$ in $dW(a\otimes b)$ are respectively $(-1{)}^{pq}$ and $(-1{)}^{pq+q}$. In $Wd(a\otimes b)$ they are $(-1{)}^{p+p(q-1)}$ and $(-1{)}^{(p-1)q}$, respectively. Each corresponding pair agrees modulo two, so $dW=Wd$. Also $W^2=1$. Terms involving $da$ or $db$ in degree zero are absent, so the calculation includes $p=0$ and $q=0$.

A shuffle path has $p$ horizontal and $q$ vertical steps. Its permutation sign is $(-1{)}^N$, where $N$ counts vertical steps occurring before horizontal steps. Interchanging the two types of steps replaces $N$ by $pq-N$, since each horizontal/vertical pair contributes to exactly one of the counts. It therefore changes the sign by $(-1{)}^{pq}$. The affine simplex of the swapped path is the original one followed by factor interchange. Matching paths bijectively in the finite shuffle sums gives ${\tau }_{\mathrm{\#}}S=SW$. This is a signed-shuffle calculation, not merely naturality for maps of the two factors.

Put $B=WA{\tau }_{\mathrm{\#}}$ and $L=WUW$. These are natural (with simultaneous maps of $X$), and step 1.2 gives $BS=WASW$. Thus $dL+Ld=W(AS-1)W=BS-1.$ The map $B$ is a chain map by step 1.1, [F1], and the fact that postcomposition commutes with face boundary. Define $K=LA-BV$. Using the two supplied homotopy equations yields $dK+Kd=(BS-1)A-B(SA-1)=B-A\mathrm{.}$ This is an explicitly specified natural homotopy.

Since $\tau \mathrm{\Delta }=\mathrm{\Delta }$, precomposing step 2.1 with the chain map ${\mathrm{\Delta }}_{\mathrm{\#}}$ gives $d(K{\mathrm{\Delta }}_{\mathrm{\#}})+(K{\mathrm{\Delta }}_{\mathrm{\#}})\mathrm{\partial }=W{D}_X-D_X\mathrm{.}$ In degree zero the twists have sign $+1$ and the maps agree on vertices. If $X$ is empty or $R=0$, all maps have zero complexes as domain and codomain. Degenerate singular simplices and one-point spaces use the same shuffle paths and homotopies; nothing was normalized away. The construction uses finite signed sums and the supplied homotopies, not a selection of new fillers, so it is choice-free.