Let $A$ be a ring, $S$ be a multiplicative subset, $M$ an $A$-module. let $\iota : A \to S^{-1}A$ be the map $a \mapsto a/1$. $\iota$ can be defined categorically as an initial object in the category of ring homomorphisms whose domain is $A$ which maps $S$ into the units (morphisms in this category are commuting triangles).
In Atiyah McDonald, as far as I can tell, there is no similar categorial definition for $S^{-1}M$. My question is: What are some good ways of defining $S^{-1}M$ categorically?