Equivalent characterizations of the essential image of reflective subcategories I'm currently trying to prove the third item of the following exercise from Category Theory in Context,

Exercise 4.5.vii. Consider a reflective subcategory inclusion $D \hookrightarrow C$ with reflector $L: C \to D$.
  
  
*
  
*Show that $\eta L = L\eta$, and that these natural transformations are isomorphisms.
  
*Show that an object $c \in C$ is in the essential image of the inclusion $D\hookrightarrow C$, meaning that it is isomorphic to an object in the subcategory $D$, if and only if $\eta_c$ is an isomorphism.
  
*Show that the essential image of $D$ consists of those objects $c$ that are local for the class of morphisms that are inverted by $L$. That is, $c$ is in the essential image if and only if the pre-composition functions
  $$
C(b, c) \xrightarrow{f^*} C(a, c)
$$
  are isomorphisms for all maps $f : a \to b$ in $C$ for which $Lf$ is an isomorphism in $D$. This explains why the reflector is also referred to as “localization.”
  

Here $\eta$ is the unit of the adjunction. I have managed to prove the first two items. However, reflective subcategories were just briefly introduced and so I am not sure how to relate item $(3)$ with $(2)$, if this is the right path to begin with. Any hints? I would also appreciate if someone can shed some light on the term localization and why $(3)$ 'explains' this nomenclature.
 A: Hint: First prove that $iLc$ satisfies the condition in 3, where $i:D\to C$ is the inclusion. Then use apply 2 to conclude. 
Regarding "localization" the point is that $L$ is characterized by the arrows it inverts, so the language is being imported from commutative algebra. Specifically, the motivating situation is that of $R[S^{-1}]$-modules, where $S$ is a multiplicative set in a commutative ring $R$. Every $R[S^{-1}]$ is functorially an $R$-module by restricting the scalars, and this functor is fully faithful, with the left adjoint $R[S^{-1}]\otimes_R (-)$. So this is a reflective subcategory. Furthermore an $R$-module admits an $R[S^{-1}]$-action if and only if it is local for those $R$-module maps inverted by tensoring with $R[S^{-1}]$, as in Riehl's point 3. The classical algebraic framework would reduce those maps to the maps $s:R\to R$ determined by elements of $S$, while the general categorical framework instead asks for locality with respect to all maps $\eta_A:A\to A[S^{-1}]$, for $A$ an $R$-module. We can bridge these frameworks by observing that locality with respect to each $s$ is equivalent to locality with respect to the single unit map $R\to R[S^{-1}]$, which implies locality with respect to every $\eta_A$ by considering the action of $L$ on a presentation of $A$.
A: Judging from the comments, I feel like that this addendum to @Kevin Arlin's answer may be useful.
Here's a proof for one direction of the third item.
Consider $\eta_c : c \to iLc$; by the first item $iL\eta_c : iLc \to iLc$ is iso so that $\eta^{*}_c: \mathsf{C}(iLc,c) \to \mathsf{C}(c,c)$ is a bijection.
In particular there is a unique $s: iLc \to c$ such that $s \eta_c = 1_c$.
This implies that $iLs \cdot iL\eta_c = 1_{iLc}$; now, $iL\eta_c=\eta_{iLc}$ is an isomorphism, and by the triangular identities, its inverse is $i\epsilon_{Lc}$.
Therefore $iLs=i\epsilon_{Lc}$ and
\begin{equation*}
    \eta_c s= iLs \cdot \eta_{iLc} = i\epsilon_{Lc} \cdot \eta_{iLc} = 1_{iLc}
\end{equation*}
where the first eqaulity follows from naturality.
