Relative cup products are natural and connector-compatible

Pullback of a cochain vanishing on $A$ vanishes on $A^{\mathrm{\prime }}$, and similarly for $B$. It also maps $Q$ to the primed quotient-functional complex. By [F4] it commutes with the cup formula on cochains. The quotient maps in [F1] commute with $f_{\mathrm{\#}}$, since all maps are induced by the same map of chains on $X^{\mathrm{\prime }}$. Their induced cohomology maps commute as well, and their inverses commute because they are isomorphisms. Transporting the cochain equality through these inverses proves relative naturality.

Restriction gives a termwise exact sequence $0\to Q\to C^{\ast }(X,B;R)\to C^{\ast }(A,A\cap B;R)\to 0$. To prove surjectivity, extend a cochain from simplices in $A$ by zero off $A$. It already vanishes on simplices in $A\cap B$, so the extension vanishes on $B$. Its kernel consists exactly of cochains vanishing on both $A$ and $B$, namely $Q$. Restriction commutes with coboundary; the extension need not. For a relative cocycle $z$, a lift $\tilde{z}$ has $\delta \tilde{z}\in Q$. Changing lifts changes this by a Q-coboundary; changing $z$ by $\delta w$ and lifting $w$ leaves the connecting class unchanged. These are exactly the lift calculations of [F2]. Composing this well-defined connecting class with [F1]'s inverse defines the displayed ${\mathrm{\partial }}_{A,B}$.

For the first identity represent $x$ by a cocycle $\alpha$ on $A$ and $y$ by a cocycle $\beta$ on $X$ vanishing on $B$. Extend $\alpha$ to $\tilde{\alpha }$ on $X$. The cochain $\tilde{\alpha }⌣\beta$ vanishes on $B$ and restricts to $\alpha ⌣i^{\ast }\beta$, so it is an admissible lift for step 1.2. Its coboundary is $\delta \tilde{\alpha }⌣\beta$ because $\delta \beta =0$. By [F2], $\delta \tilde{\alpha }$ represents ${\mathrm{\partial }}_{A}x$ and vanishes on $A$. Hence the very same Q-cocycle computes both sides after applying [F1]'s comparison.

For the second identity represent $x$ by a cocycle $\alpha$ on $X$ vanishing on $B$, and $y$ by a cocycle $\beta$ on $A$. Extend $\beta$ to $\tilde{\beta }$ on $X$. Now $\alpha ⌣\tilde{\beta }$ is a lift vanishing on $B$, and its coboundary is $(-1{)}^p\alpha ⌣\delta \tilde{\beta }$, because $\delta \alpha =0$. The last factor represents ${\mathrm{\partial }}_{A}y$ and vanishes on $A$. Transporting this Q-cocycle gives precisely the asserted sign and factor order.

If $B=\mathrm{\varnothing }$, the restriction sequence in step 1.2 is the pair sequence and [F1]'s comparison is the identity. If $A=\mathrm{\varnothing }$, its right-hand complex is zero and both formulas have zero sides. If $A=X$, every ${\mathrm{\partial }}_A$ and the displayed relative target are zero. For $p=0$ the second sign is positive, and degree-zero cocycles/lifts require no negative cochain. Zero ring, empty space and zero inputs give zero identities. On point spaces these are the empty/full cases; degenerate simplices are included in each extension rule. All extensions can be the stated zero extensions, so no AC or family of arbitrary lift choices is used.