Given a category $\mathcal{C}$ and a set (let's not bother with size issues here) $\mathcal{W} \subseteq \text{Mor}(\mathcal{C})$ we may form the category $\mathcal{C}[\mathcal{W}^{-1}]$ obtained by formally inverting all arrows belonging to $\mathcal{W}$. If we're lucky, then the localization functor $j: \mathcal{C} \longrightarrow \mathcal{C}[\mathcal{W}^{-1}]$ admits a fully faithful right adjoint $\iota: \mathcal{C}[\mathcal{W}^{-1}] \hookrightarrow \mathcal{C}$, in which case one speaks of a *reflective localization*.


The category $\mathcal{C}[\mathcal{W}^{-1}]$ viewed as a full subcategory of $\mathcal{C}$ via $\iota$ obviously consists of $\mathcal{W}$-local objects, i.e. for every $A \in \mathcal{C}[\mathcal{W}^{-1}]$ and every $f \in \mathcal{W}$ the map $\text{Hom}(f,A)$ is a bijection.


The nLab entry for "reflective localization" seems to claim that also the converse holds, i.e. that every $\mathcal{W}$-local object lies in the essential image of $\iota$, or equivalently, that for every $\mathcal{W}$-local $X$ the unit


$\eta(X): X \longrightarrow \iota(j(X))$


of the adjunction is an isomorphism. However I don't see how to prove that. By the triangle equation it follows (since $\iota$ is fully faithful) that $\eta(X)$ becomes an isomorphism after applying $j$, but I don't see how this helps in proving that $\eta(X)$ itself is an isomorphism.


So the question is, does this reverse implication hold at all?



Reflective Localizations vs. categories of local objects
To avoid confusing myself, I will write $L : \mathcal{C} \to \mathcal{C} [\mathcal{W}^{-1}]$ for the localising functor and $R : \mathcal{C} [\mathcal{W}^{-1}] \to \mathcal{C}$ for its right adjoint. (Note that $R$ is *automatically* fully faithful – the hard part is existence!) 


As you say, for any object $Y$ in $\mathcal{C} [\mathcal{W}^{-1}]$, $R Y$ is automatically a $\mathcal{W}$-local object in $\mathcal{C}$: indeed, for any object $Z$ in $\mathcal{C}$, $Z$ is a $\mathcal{W}$-local object if and only if $\mathcal{C} (-, Z) : \mathcal{C}^\mathrm{op} \to \mathbf{Set}$ factors through $L : \mathcal{C} \to \mathcal{C} [\mathcal{W}^{-1}]$. So suppose $Z$ is a $\mathcal{W}$-local object. Since $\epsilon : L R \Rightarrow \mathrm{id}\_{\mathcal{C} [\mathcal{W}^{-1}]}$ is a natural isomorphism, the triangle identities imply $L \eta : L \Rightarrow L R L$ is also a natural isomorphism. In particular,
$$\mathcal{C} (\eta\_Z, Z) : \mathcal{C} (R L Z, Z) \to \mathcal{C} (Z, Z)$$
is a bijection, so there is a unique morphism $\alpha : R L Z \to Z$ such that $\alpha \circ \eta\_Z = \mathrm{id}\_Z$. But $L \alpha = \epsilon\_{L Z}$, so
$$\eta\_Z \circ \alpha = R L \alpha \circ \eta\_{R L Z} = R \epsilon\_{L Z} \circ \eta\_{R L Z} = \mathrm{id}\_{R L Z}$$
and therefore $\eta\_Z : Z \to R L Z$ is indeed an isomorphism.


So the conclusion is that $\mathcal{C} [\mathcal{W}^{-1}]$ is equivalent to the full subcategory of $\mathcal{C}$ spanned by the $\mathcal{W}$-local objects.


