relative singular homology functor

Given the category of topological pairs and \(n \in \mathbb{N}_0\), the relative singular homology functor is defined as functor

\begin{align*} H_n(-,-): \mathrm{TopPairs} \rightarrow& \mathrm{Ab} \\ (X,A) \mapsto& H_n(C^{\mathrm{sing}}(X,A) \\ (f: (X,A) \rightarrow (Y,B)) \mapsto& H_n(f': (C^{\mathrm{sing}}(X,A)) \rightarrow (C^{\mathrm{sing}}(Y,B)) \end{align*}

where \(H_n: \mathrm{Ch_*}(\mathrm{Ab}) \rightarrow \mathrm{Ab}\) is the homology functor and \(C^{\mathrm{sing}}(X,A)\) is the relative singular chain complex functor