The simplicial-to-singular chain map commutes with boundaries

If $n=0$, then both ${\mathrm{\partial }}^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$ and ${\mathrm{\partial }}^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}$ are zero maps in degree $0$, so the degree-zero claim is immediate.

Assume $n\ge 1$ and let $\sigma =[v_0,\dots ,v_n]$ be the increasing representative of an oriented simplex of $K$. By [L1], the singular chain ${\mathrm{\Phi }}_n(\sigma )$ is the affine characteristic simplex ${\chi }_{[v_0,\dots ,v_n]}$. Restricting along the face map ${\delta }_i$ therefore produces the affine characteristic simplex of the face $[v_0,\dots ,{\widehat{v}}_i,\dots ,v_n]$, listed in the induced increasing order.

Applying [L2] and [L3] to step 1.2 gives ${\mathrm{\partial }}^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}{\mathrm{\Phi }}_n(\sigma )={\sum }_{i=0}^n(-1{)}^i{\mathrm{\Phi }}_{n-1}([v_0,\dots ,{\widehat{v}}_i,\dots ,v_n])={\mathrm{\Phi }}_{n-1}{\mathrm{\partial }}^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(\sigma )$ for the increasing representative. The same overall orientation sign $\epsilon (\tau )$ from [L1] multiplies both sides for an arbitrary oriented simplex $\tau$, so the identity holds on every generator and hence on all simplicial chains by linearity; step 1.1 handles degree $0$.