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.