Finite dimensional skeletal exactness computes axiomatic homology

Use the filtration and notation of F1. Write $E_q^r=h_q(F_r,A)$. The triple sequence and concentration of $h_q(F_r,F_{r-1})$ in degree $r$ give $E_q^r=0$ for $q<0$ or $q>r$, inductively from $E_q^{-1}=0$. They also give $E_q^{r-1}\to E_q^r$ an isomorphism for $q<r-1$, and a surjection for $q=r-1$. Since the filtration terminates, $E_n^{n+1}\cong h_n(X,A)$, taking $F_r=X$ above the top dimension.

For $n\ge 0$, ${\rho }_n:E_n^n\to C_n^h$ is injective since $E_n^{n-1}=0$. Likewise ${\rho }_{n-1}:E_{n-1}^{n-1}\to C_{n-1}^h$ is injective when $n\ge 1$. Exactness then yields ${\rho }_n(E_n^n)=\ker d_n$; for $n=0$ this says $E_0^0=C_0^h$ since $E_{\ast }^{-1}=0$.

The surjection $i:E_n^n\to E_n^{n+1}$ has kernel $\mathrm{im}{\delta }_{n+1}$. Since ${\rho }_n{\delta }_{n+1}=d_{n+1}$, define $\alpha (x)=[{\rho }_n(y)]$ for any lift $i(y)=x$. Two lifts differ by a ${\delta }_{n+1}$ image, so the class is independent. Every cycle is ${\rho }_n(y)$ by the preceding step, giving surjectivity. If ${\rho }_n(y)=d_{n+1}(z)$, injectivity of ${\rho }_n$ implies $y={\delta }_{n+1}(z)$ and hence $x=0$, proving injectivity.

Every map of the filtered exact diagrams carries a lift to a lift and commutes with $\rho ,\delta ,i$, so it commutes with $\alpha$. This proves cellular naturality and uniqueness of the isomorphism defined by the lift rule. For pair boundaries use the same construction on the cofiber sequence $A_{+}\to X_{+}\to X_{+}\mathrm{/}{A}_{+}\to \mathrm{\Sigma }{A}_{+}$. Give each suspension cell the cone orientation. Its cellular chain group in degree $n+1$ is the reduced degree-$n$ group of $A_{+}$, and its boundary is the suspended boundary with the cone sign convention. The cofiber map sends a relative cellular cycle represented by a chain $c$ of $X$ to the suspended class of $dc\in C_{n-1}^h(A)$: the faces outside $A$ cancel because $c$ was a relative cycle. In the exact diagram this is exactly the triple connecting map. Desuspending therefore identifies the axiomatic pair boundary with the chain connecting map $[c]\mapsto [dc]$. The lift construction commutes with suspension because its $\rho$ and $\delta$ maps do.

The direct-sum decomposition by relative cells splits $0\to C_{\ast }^h(A)\to C_{\ast }^h(X)\to C_{\ast }^h(X,A)\to 0$ degreewise, so the preceding chain-boundary formula is defined for arbitrary $G$, without a flatness assumption. Negative degrees are zero by the first step. Empty pairs and pairs with no relative cells give zero on both sides.