I'm learning about Bousfield localizations. For a triangulated category satisfying some axioms, a Bousfield localizations can be described as an idempotent functor $L:D \to D$.

I thought there is a bijection between Bousfield localizations, and localizing subcategories (the kernel of $L$ determines $L$).

But there is a *set* of localizing subcategories of $D$, and a *category* of idempotent functors $D \to D$. I guess what I mean is, the idempotent functor can have automorphisms (even the identity functor can have automorphisms)

Is there a way to fix that, in the definition of the localization functor $L:D \to D$? To "rigidify" i.e. add a little more data so that it has no automorphisms?

The next thing I want to understand is, why do Bousfield localizations form a poset, but I got hung up on this. If I am thinking about it the wrong way, let me have it!