Let $R$ be a commutative ring with unit and let $S \subseteq R$ be a multiplicative closed subset. Consider the ring of fractions $S^{-1}R$ with the homomorphism $f\colon R \to S^{-1}R$ which sends $r$ to $(rs)/s$. If $T$ is a commutative ring with unit for which there exists a homomorphism $\alpha$ s.t. $\alpha(s)$ is invertible in $T$ for any $s$ in $S$, then there exists a unique homomorphism $g\colon S^{-1}R \to T$. This happens for all $T$ and for all $\alpha$ satisfying the previous properties.
Now, let $A$ be a commutative ring with unit such that there exists a homomorphism $\beta\colon R \to A$ with $\beta(s)$ invertible in $T$ for every $s \in S$. Suppose that for all commutative rings with unit $T$ and for all homomorphisms $\alpha\colon R \to T$ with $\alpha(s)$ invertible in $T$ for all $s\in S$ there exists a unique homomorphism $\gamma\colon A \to T$ such that $\gamma \circ \beta = \alpha$, then
is it true that there exists an isomorphism between $S^{-1}R$ and $A$ ?