Let $R$ be an integral domain $M$ be $R$-module. $S^{-1}M$ is localization of $M$ at $S$ as the $S^{-1}R$-module
$i_M:M\rightarrow S^{-1}M$ ; $m\mapsto (m,1)= m/1$
I showed that this is an $R$-module homomorphism.
Let $f: M\rightarrow N$ be an $R$-module homomorphism. Show that there is a unıque $S^{-1}R$-module homomorphism $g: S^{-1}M \rightarrow S^{-1}N$ such that
$i_N \circ f=g \circ i_M$
Uniqueness part is solvable for me, but I can not prove existence part