I am trying to show that $S^{-1}R=\operatorname{colim}F(s)$, where $S$ is a multiplicative closed set in a commutative ring $R$ and $F$ is a functor from a filtered category $I$ to mod-$R$ and $I$ is constructed as follows: objects of $I$ are just the elements of $S$ and the morphisms are $$\operatorname{Hom}\_{I}(s\_{1},s\_{2})=\{s \in S : s\_{1}s=s\_{2}\}$$
and $F$ is defined as follows: it takes $s$ to $F(s)=R$ and a morphism $s\_{1}\rightarrow s\_{2}$ to the multiplication by $s$ from $R\rightarrow R$.

I tried it to this way: firstly I showed that $S^{-1}R$ is co-cone with the morphism $F(s)\rightarrow S^{-1}R$ taking $1$ to $1/s$. Then I want to show that it satisfies universal property of colim. Which I am not getting.

Thanks in advance.