Localization of modules as adjunction

Localization of modules as adjunction

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 ?



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.