Editing Localization of a category (section)

===Alternative definition===

Some authors also define a ''localization'' of a category ''C'' to be an [[idempotent]] and coaugmented functor. A coaugmented functor is a pair ''(L,l)'' where ''L:C → C'' is an [[endofunctor]] and ''l:Id → L'' is a natural transformation from the identity functor to ''L'' (called the coaugmentation). A coaugmented functor is idempotent if, for every ''X'', both maps ''L(l<sub>X</sub>),l<sub>L(X)</sub>:L(X) → LL(X)'' are isomorphisms. It can be proven that in this case, both maps are equal.<ref>[http://www.math.uchicago.edu/~mitya/idempotents.pdf Idempotents in Monoidal Categories]</ref>

This definition is related to the one given above as follows: applying the first definition, there is, in many situations, not only a canonical functor <math>C \to C[W^{-1}]</math>, but also a functor in the opposite direction,
:<math>C[W^{-1}] \to C.</math>
For example, modules over the localization <math>R[S^{-1}]</math> of a ring are also modules over ''R'' itself, giving a functor
:<math>\text{Mod}_{R[S^{-1}]} \to \text{Mod}_R </math>
In this case, the composition
:<math>L : C \to C[W^{-1}] \to C</math>
is a localization of ''C'' in the sense of an idempotent and coaugmented functor.