In notes about Algebraic Number Theory, there is the following exercise: Let $K$ be a number field with ring of integers $R$, and let $S$ be a finite subset of nonzero prime ideals of $R$. Show that there is a canonical exact sequence of abelian groups $$1 \rightarrow R^* \rightarrow \oplus_{\mathfrak{p}\in S} (K^*/R^*_\mathfrak{p})\rightarrow \text{Cl}(R) \rightarrow \text{Cl}(R^S) \rightarrow 1$$
I am having trouble in finding the map $\psi: \oplus_{\mathfrak{p}\in S} (K^*/R^*_\mathfrak{p})\longrightarrow \text{Cl}(R)$.
I have tried taking $(u_\mathfrak{p}R_\mathfrak{p})_{\mathfrak{p} \in S} \in \oplus_{\mathfrak{p}\in S} (K^*/R^*_\mathfrak{p})$ to $[\prod_\mathfrak{p} u_\mathfrak{p}R^S]$ for example, or to $[\sum_\mathfrak{p} u_\mathfrak{p}R^S]$, but in the first case the homomorphism is trivial and in the second case the map doesn't seem to be a homomorphism.
Any thoughts?