A question on a problem on localization from Atiyah (3.8) I was having trouble with the following problem from Chapter $3$ of Atiyah-MacDonald
Let $S, T$ be multiplicatively closed sets in the ring $A$, such that $S\subseteq T$.
Let $\varphi : S^{−1}A \to T^{−1}A$ be the homomorphism which maps each $r/s \in S^{−1}A$ to $r/s$  viewed as an element of $T^{−1}A$. Show that the following are equivalent:

(i) $\varphi$ is a bijection;
(ii) $t/1$ is a unit in $S^{−1}A$ for all $t\in T$;

My Attempts
I am assuming $\varphi$ is bijective
I first tried to show this directly by noting that since $\varphi$ is bijective we have that $\varphi(a/s)=t/1$   .
This means we have that $a/s=t/1$ and therefore there is a $w \in T$ where $w(a-st)=0$. I am stuck at this point.
Another approach, I saw recommended was to consider to universal property associated to localization. I am confused on how to apply it, do I consider this with respect to $A \rightarrow T^{-1}A$ where $S$ is the closed mult. set being considered in which case there should be a map from $S^{-1}A$ to $T^{-1}A$. I'm not sure how that relates to $\varphi$ being bijective.
Any help would be appreciated.

$t/1$ is a unit in $T^{-1}R$.  Because $\varphi$ is an isomorphism, $t/1$ is a unit in $S^{-1}R$.
