In the paper, Topologically Defined Classes of Commutative Rings, localization of the pullback diagram (with $v,$ surjective)
$$
\begin{array}
DD & \stackrel{v\ '}{\longrightarrow} & A \\
\downarrow{u' } & & \downarrow{u} \\
B & \stackrel{v}{\longrightarrow} & C
\end{array}
$$
is given as Proposition 1.9:

... Conversely, if $S_A$ is a multiplicatively closed set of $A$ and if $S_B$ is a multiplicatively closed of $B$ and if $u(S_A)=v(S_B)=S_C$ then $S_A^{-1}A \times_{S_C^{-1}C}S_B^{-1}B \cong (S_A \times_{S_C}S_B)^{-1} D.$

I can not see how a typical element of one side mapped to the other side by this isomorphism.

Can you help please?

Thank you.