Two homology theories with different coefficient groups

Given: The objects and hypotheses in the statement above.

For any abelian group $G$, $H_0(\ast ;G)\cong G$ and $H_n(\ast ;G)=0$ for every integer $n\mathrm{\ne }0$. For every set-indexed family of pairs the canonical map ${⨁}_{\alpha }{H}_n(X_{\alpha },A_{\alpha };G)⟶H_n({⨆}_{\alpha }{X}_{\alpha },{⨆}_{\alpha }{A}_{\alpha };G)$ is an isomorphism. Together with the structural axioms, singular homology is an ordinary theory with coefficient group $G$. (Singular homology satisfies dimension and arbitrary additivity)

For any two ordinary homology theories $h,k$ on all CW pairs and a specified coefficient isomorphism $u:h_0(\ast )\to k_0(\ast )$, there is a unique natural equivalence $h\to k$ normalized by $u$ and commuting with connecting homomorphisms. In particular, a theory with coefficient group $G$ is naturally equivalent to singular homology with coefficients $G$, normalized by ${\mathrm{i}\mathrm{d}}_G$. Arbitrary additivity is part of the hypotheses. (Eilenberg steenrod uniqueness on all cw pairs)