This is probably a dumb question but this is going over my head at the moment, I came here from nlab's entry on localization (http://ncatlab.org/nlab/show/localization).
Let $C$ be a category, let $W \in Mor(C)$ be a collection of morphisms, let $L_WC$ be the reflective localization of $C$. Are $C$ and and $L_WC$ equivalent categories? The inclusion functor $L_WC \hookrightarrow C$ is fully faithful and $L_WC$ has the same objects as $C$, what am I missing?
$C$ and $L_W C$ having the same objects is a red herring and an artifact of the construction of $L_W C$. In particular, the "inclusion functor" won't be the identity on objects. Being fully faithful, it embeds $L_W C$ into $C$ in the categorical sense: it establishes an equivalence between $L_W C$ and a full subcategory of $C$, in this case the subcategory of $W$-local objects (proof).