Localization of modules as adjunction

Question

Usually, the localization of a $R$-module $M$ by a multiplicative subset $S \subseteq R$ with $1 \in S$ is categorically defined as the initial object of the full subcategory $\mathbf C$ of $M \, \backslash \, R\!-\!\mathbf{Mods}$ constituted by the objects $M \to N$ with $N$ such that $s\times \cdot \in \mathrm{Aut}_{R-\mathbf{Mods}}(N)$ for all $s\in S$.

I was wondering if the following definition is valid as well : let $\mathbf C$ be the full subcategory of $R\!-\!\mathbf{Mods}$ constituted by the objects $N$ such that $s\times \cdot \in \mathrm{Aut}_{R-\mathbf{Mods}}(N)$ for all $s\in S$ ; then the functor of localization $S^{-1}$ is the left adjoint of the functor of inclusion $$ i \colon \mathbf C \to R\!-\!\mathbf{Mods}.$$ It would have the advantage to present $S^{-1}$ directly as a (colimits preserving) functor. The canonical homomorphisms $M \to S^{-1}M$ then are just the (components of the) unit of the adjunction $S^{-1} \dashv i$.

However, I never saw such a presentation of the localization. Is my statement wrong ?

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P.S. : I am aware that in either case, I need to explicitly construct the localization. My question is really about the point of view on the localization.

Answer 1

This is an instance of a standard fact about left adjoints:

Let $U : \mathcal{C} \to \mathcal{D}$ be a functor. Then $U$ has a left adjoint if and only if each comma category $(D \downarrow U)$ has an initial object, and the value of the left adjoint at an object $D$ is an initial object in $(D \downarrow U)$.