Non-Naturality of the Splitting in the Universal Coefficient Formula

I want to show the non-naturality of the splitting in the universal coefficient formula for homology. The s.e.s. is $$0\to H_q(X,X';R)\otimes_R N\to H_q(X,X';N)\to Tor^R_1(H_{q-1}(X,X';R),N)\to 0$$ where $R$ is a PID and $N$ an $R-$module.

I found a counter example with the map $\mathbb{RP}^2\to S^2$ that collapses the 1-cell of $\mathbb{RP}^2$ to a point, but I don't get why this map works.

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