Naturality of the singular cohomology pair sequence

If a cochain $\phi$ on $Y$ vanishes on simplices in $B$, then $\phi f_{\mathrm{\#}}$ vanishes on simplices in $A$, since their composites have image in $B$. Thus the cochain pullback of [F2] restricts to $C^{\ast }(Y,B;G)\to C^{\ast }(X,A;G)$. It commutes with the differential and so preserves relative cocycles and coboundaries, inducing $f^{\ast }$ by [F3]. The literal composition and identity formulas from [F2] restrict to these subcomplexes, hence hold on their quotient cohomology.

The square with relative inclusion commutes because both composites send $\phi$ to $\phi f_{\mathrm{\#}}$ viewed as an absolute cochain. The restriction square commutes since restricting $\phi f_{\mathrm{\#}}$ to a simplex $\sigma$ in $A$ gives $\phi (f\sigma )$, also the value of $(f{\mathrm{\mid }}_A{)}^{\ast }(\phi {\mathrm{\mid }}_B)$ on $\sigma$. Equality on simplices is equality of cochains by linear extension. These squares descend to cohomology.

Coefficient postcomposition preserves zero values on subspace chains and commutes with coboundaries by [F2]. Thus it gives maps on all three kinds of cohomology. It commutes with inclusions and restrictions by the pointwise definitions. If $\tilde{a}$ extends $a$, then $u\tilde{a}$ extends $ua$ and $\delta (u\tilde{a})=u\delta \tilde{a}$; therefore it also commutes with the connector of [F1]. Finally $u(\phi f_{\mathrm{\#}})=(u\phi )f_{\mathrm{\#}}$ proves commutation with pair pullback, and coefficient identity/composition laws restrict from [F2].

For a cocycle $a$ on $B$, let $\tilde{a}$ be any extension to $Y$, available by [F1]. The cochain $\tilde{a}{f}_{\mathrm{\#}}$ extends $(f{\mathrm{\mid }}_A{)}^{\mathrm{♯}}a$ on $A$. Its differential is $(\delta \tilde{a})f_{\mathrm{\#}}$ by [F2]. Hence both sides of the asserted connector square are represented by this same relative cocycle. Independence from the extension is precisely [F1], so the square commutes on cohomology. It is not necessary that pullback preserve the particular extension-by-zero section.

Steps 1.1, 1.2, 2.1 and 1.3 prove every square and both functorialities. In degree zero the connector still uses an extended zero-cocycle and the same calculation; negative groups and their maps are zero. For $A=\mathrm{\varnothing }$ or $A=X$, and likewise for $B$, the inclusion/restriction identities agree with the endpoint sequences of [F1] whenever $f$ is a map of pairs. Empty $X$, point spaces and zero coefficients are covered by these literal formulas. No global family of extensions is chosen: one may use the specified zero extension for the one cocycle under consideration, so no AC is used.