Naturality of singular Mayer–Vietoris

Composition with $f$ commutes with the singular boundary, since $(f\circ \sigma )\circ {\delta }_j=f\circ (\sigma \circ {\delta }_j)$. The cover conditions give chain maps on the overlap, on each summand, and $f_s:A_X\to A_Y$. They commute with $i(c)=(c,-c)$ and $j(u,v)=u+v$ by additivity. Thus they give a morphism of the short exact sequences in Short exact chain Mayer–Vietoris sequence, and The long exact homology sequence is natural gives a commuting ladder of their homology sequences.

Let ${\iota }_X:A_X\hookrightarrow C_{\bullet }(X;G)$ and ${\iota }_Y:A_Y\hookrightarrow C_{\bullet }(Y;G)$ be the inclusions. On every small simplex both composites are the same singular simplex $f\circ \sigma$, so $f_{\mathrm{\#}}{\iota }_X={\iota }_Y{f}_s$. The maps $H_n({\iota }_X),H_n({\iota }_Y)$ are isomorphisms by Cover-small chains compute singular homology. Therefore $H_n(f_s)H_n({\iota }_X{)}^{-1}=H_n({\iota }_Y{)}^{-1}{H}_n(f)$. Transporting the ladder of step 1.1 along these isomorphisms yields exactly the ordinary homology maps and sequences of Mayer–Vietoris sequence in singular homology.

In particular, for a small cycle $z=u+v$, its image is $f_{\mathrm{\#}}u+f_{\mathrm{\#}}v$ and $f_{\mathrm{\#}}(\mathrm{\partial }u)=\mathrm{\partial }{f}_{\mathrm{\#}}u$. Thus the connector sends the image class to the image of $[\mathrm{\partial }u]$, with the same positive U-boundary sign; independence of the chosen small representative and decomposition is supplied by Well-definedness of the Mayer–Vietoris connector and the inclusion isomorphisms. This proves every connecting square as well as the ordinary squares. The argument includes degree zero, the terminal maps to zero, empty overlap or empty cover members, and $G=0$.