Let $A$ be a ring and $S\subset A$ be a multiplicatively closed system and $M$ an $A$-module. I have seen the following universal property of the localisation of $M$ at $S$:
$S^{-1}M$ is an $A$-module map $M \rightarrow S^{-1}M$ which is an initial object in the following category:
- the objects of the category are $A$-module maps $M \rightarrow N$, where $N$ is also an $S^{-1}A$-module with the $A$-module structure given by multiplying with $a/1$;
- the morphisms between two objects $\phi_1: M \rightarrow N_1$ and $\phi_2: M \rightarrow N_2$ are module maps $\alpha: N_1 \rightarrow N_2$ satisfying $\alpha \circ \phi_1 = \phi_2$.
My question is, in this definition, do we suppose $\alpha$ to be $A$-module maps, or do we put the stronger condition that $\alpha$ are $S^{-1}A$-module maps.
While it seems natural to put the stronger condition on the $\alpha$'s, is there any concrete reason to put the stronger condition other than the fact that... we can (in the sense that the well-known concrete construction for localization does guarantee an $S^{-1}A$-module map between $S^{-1}M$ and any $S^{-1}A$ module $N$)?