Let $A$ be a commutative ring and $S$ a multiplicatively closed system. I am trying to see that if $\alpha \colon N_1 \rightarrow N_2$ is an $A$-linear map of $A$-modules which are in fact $S^{-1}A$-modules, then $\alpha$ is in fact $S^{-1}A$-linear. I want to do this just using the universal property of the localisation. Is this possible?
My attempt is as follows: $\alpha$ being $A$-linear just means that we have a commuting diagram
Now by the universal property, $A \rightarrow S^{-1}A$ is initial among ring maps $A \rightarrow B$ sending $S$ to invertibles. Now since $N_1$, $N_2$ are even $S^{-1}A$ modules, $A \rightarrow \mathrm{End}(N_1)$ and $A \rightarrow \mathrm{End}(N_2)$ are two maps sending $S$ to invertibles. Hence, we have a larger diagram which I want to show is commutative:
The problem is, how can I conclude from this that $$ S^{-1}A \rightarrow \mathrm{End}(N_1) \rightarrow \mathrm{Hom}(N_1, N_2) = S^{-1}A \rightarrow \mathrm{End}(N_2) \rightarrow \mathrm{Hom}(N_1, N_2) \,. $$ One would be tempted to use the uniqueness of the map from $S^{-1}A$ to any ring $B$ where $A \rightarrow B$ sends $S$ to invertibles. The problem is that $\mathrm{Hom}(N_1, N_2)$ is not a ring!
Even so, I haven't used uniqueness yet, but I don't see how.
Let us abbreviate $\mathrm{Hom}_ℤ$ and $\mathrm{End}_ℤ$ as $\mathrm{Hom}$ and $\mathrm{End}$ respectively.
Framework
Let us first consider two abelian groups $M$ and $N$. The abelian group $\mathrm{Hom}(M, N)$ becomes a left $\mathrm{End}(N)$-module via postcomposition and a right $\mathrm{End}(M)$-module via precomposition. For a given element $f$ of $\mathrm{Hom}(M, N)$ we can consider the set $$ E_f := \{ (h, k) ∈ \mathrm{End}(M) × \mathrm{End}(N) \mid h f = f k \} \,. $$ This set is a subring of $\mathrm{End}(M) × \mathrm{End}(N)$. It is also closed under invertibility, in the following sense: whenever $(h, g)$ is an element of $E_f$ that is invertible in $\mathrm{End}(M) × \mathrm{End}(N)$, its inverse $(h, k)^{-1} = (h^{-1}, k^{-1})$ is again contained in $E_f$.
Observation
The ring $E_f$ helps us understand module structures on $M$ and $N$ for which $f$ is a homomorphism.
Let $B$ be a ring. A $B$-module structure on $M$ is the same a homomorphism of rings from $B$ to $\mathrm{End}(M)$. Simultaneous $B$-module structures on $M$ and $N$ are therefore the same as a pair $(φ_1, φ_2)$ of homomorphisms of rings $φ_1 \colon B \to \mathrm{End}(M)$ and $φ_2 \colon B \to \mathrm{End}(N)$. Such a pair of homomorphisms is the same as a single homomorphism $φ \colon B \to \mathrm{End}(M) × \mathrm{End}(N)$ (whose components are given by $φ_1$ and $φ_2$).
The map $f$ is a homomorphism of $B$-modules if and only if the image of $φ$ is contained in the subring $E_f$ of $\mathrm{End}(M) × \mathrm{End}(N)$. In other words, if and only if there exists a homomorphism of rings $φ' \colon B \to E_f$ such that $φ = j ∘ φ'$, where $j$ is the inclusion map from $E_f$ to $\mathrm{End}(M) × \mathrm{End}(N)$.
The problem at hand
Let us now consider the original problem: $M$ and $N$ are not just abelian groups, but $S^{-1} A$-modules. These simultaneous $S^{-1} A$-module structures correspond to a homomorphism of rings $φ \colon S^{-1} A \to \mathrm{End}(M) × \mathrm{End}(N)$. We can regard $M$ and $N$ as $A$-modules by restriction, and these simultaneous $A$-module structures correspond to a homomorphism of rings $ψ \colon A \to \mathrm{End} × \mathrm{End}(N)$. The homomorphism $ψ$ is just the restriction of $φ$ to $A$. More precisely, $ψ = φ ∘ i$ where $i$ is the canonical homomorphism from $A$ to $S^{-1} A$.
Suppose that $f$ is not only additive, but also $A$-linear. This means that the image of $ψ$ is contained in $E_f$. In other words, there exists a homomorphism of rings $ψ'$ from $A$ to $E_f$ with $φ' = j ∘ ψ'$.
For every element $s$ of $S$ the element $ψ'(s)$ of $E_f$ is invertible in $\mathrm{End}(M) × \mathrm{End}(N)$, because $s / 1$ is invertible in $S^{-1} A$ and $ψ'(s) = ψ(s) = φ(s / 1)$. We have seen that $E_f$ is closed under invertibility, so it follows that $ψ'(s)$ is already invertible in $E_f$. It now follows from the universal property of the localization $S^{-1} A$ that there exists a unique homomorphism of rings $φ'$ from $S^{-1} A$ to $E_f$ with $φ' ∘ i = ψ'$. We have $j ∘ φ' ∘ i = j ∘ ψ' = ψ = φ ∘ i$ and therefore $j ∘ φ' = φ$. In other word, $φ'$ is the restriction of $φ$ to a homomorphism of rings from $S^{-1} A$ to $E_f$. The existence of this restriction tells us that $f$ is $S^{-1} A$-linear.