Let $S$ be a class of maps of a category $\mathcal{C}$. The localization of $\mathcal{C}$ at $S$ is reflective when the localization functor $\mathcal{C} \to \mathcal{C}[S^{-1}]$ has a fully faithful right adjoint.

**Question 1:**

Are there any general conditions on $S$ which imply that $\mathcal{C}[S^{-1}]$ is reflective?

Suppose that $\mathcal{C}$ is locally presentable and that $S$ is a small set. Then the full subcategory $\mathcal{C}$ consisting of $S$-local objects is reflective. Let $\widetilde{S}$ be the class of $S$-local maps. Then $S \subseteq \widetilde{S}$ and $\mathcal{C}[S^{-1}]$ is reflective if and only if $\mathcal{C}[S^{-1}] = \mathcal{C}[\widetilde{S}^{-1}]$.

**Question 2:**

What are natural/useful examples of non-reflective localizations $\mathcal{C}[S^{-1}]$ (where $\mathcal{C}$ is locally presentable and $S$ is a set)?

The localization functor $\mathcal{C} \to \mathcal{C}[\widetilde{S}^{-1}]$ satisfies a modified version of the universal property of $\mathcal{C}[S^{-1}]$. Namely, we require all functors in the definition of a localization to be left adjoints. This definition makes sense for an arbitrary category $\mathcal{C}$ and an arbitrary class of maps $S$.

**Question 3:**

Is there a name for functors satisfying this universal property? Does it appear in the literature?

I want to call such a functor a *reflective localization* of $\mathcal{C}$ at $S$, but this terminology clashes with the ordinary notion of a reflective localization. Let us call it a *quasireflective localization* at $S$. Every reflective localization at $S$ is a quasireflective localization at $S$ by this proposition, but a quasireflective localization at $S$ (if it exists) is a reflective localization at $\widetilde{S}$ rather than at $S$. Thus, we can ask a weaker version of Question 2:

**Question 2':**

What are examples of quasireflective localizations at $S$ which are not reflective localizations at $S$.