This is cross-posted from MSE (and substantially re-written) after receiving no answers.

Suppose $\mathcal C$ is a category and $S \subseteq \operatorname{Mor}(\mathcal C)$ is some collection of morphisms in $\mathcal C$. Let $\operatorname{Loc}(S) $ be the full subcategory of $\mathcal C$ on the $S$-local objects. Then we have $$ \operatorname{Loc}(S) \stackrel{\iota}{\hookrightarrow} \mathcal C \stackrel{P}{\twoheadrightarrow} \mathcal C[S^{-1}], $$ where $\iota$ is the inclusion and $P$ is the canonical functor to the localization of $\mathcal C$ by $S$. We can ask about two possibly-existing objects:

- a left adjoint $L$ to $\iota$;
- a right adjoint $r$ to $P$.

In fact, (2) "subsumes" (1), in the following sense: If such an $r$ exists, then there is an equivalence of categories $j : \mathcal C[S^{-1}] \to \operatorname{Loc}(S)$ such that $r \cong \iota \circ j$. This implies that $j \circ P \vdash \iota$, so we have our $L$. Suppressing $j$, we can summarize as "if $P$ has a right adjoint, it must be $\iota$" or "if (2) exists, it is $\iota$, and hence (1) does too, and is $P$".

It's a bit tempting (at least to me) to try to extract something symmetrical like "if (2) exists, it must be $\iota$, and if (1) exists, it must be $P$" from this last summary, or maybe even "(1) exists iff (2) does", but neither of these actually follows from the statements above—to show them, we'd need a separate result about (1) "subsuming" (2) in the manner above.

So I'd like to know: **Are the following equivalent statements true?**

- If $L \vdash \iota$, then there is an equivalence of categories $j : \mathcal C[S^{-1}] \to \operatorname{Loc}(S)$ such that $L \cong j \circ P$.
- If $\iota$ has a left adjoint, then $P$ has a right adjoint.
- If $L \vdash \iota$, then $L$ exhibits $\operatorname{Loc}(S)$ as a localization of $\mathcal C$ by $S$.

I suspect the answer is "not necessarily", because the nLab page on reflective localizations lists two facts which would clearly be implied by the third statement above but which do not clearly imply it. In particular, it says that $L$ *will* exhibit $\operatorname{Loc}(S)$ as a localization of $\mathcal C$ by the class of morphisms $L$ inverts—but these morphisms are in general a strict superclass of $S$. It also says that $L$ will have a universal property *similar* to the one for a localization by $S$, but which only applies to *left adjoint* functors that invert $S$.