Uniqueness of homomorphism -universal property

Question

If $\varphi: R\to A$ is a homomorphism such that for a subset $S$, $\varphi(s)$ is inevertable for every $s\in S$ then there is a unique homomorphism $\varphi^\prime:S^{-1}R\to A$ such that $\varphi = \varphi^{\prime}\circ \pi $. where $\pi:R\to S^{-1}R$ which is defined by: $\pi(r)=\frac{r}{1}$

My problem is to prove the uniqueness of the homomorphism. Can someone give me a hint?

Answer 1

For any ring map $ψ \colon S^{-1}R → A$ with $φ=ψ∘π$ and given $r ∈ R$, $s ∈ S$, what is $ψ(\frac r 1)$ mapped to and then what about $ψ(s)ψ(\frac r s)$?