Characterizations of categorical Bousfield localizations

Proposition 21.8.9 (cf. [Lurie (2009), Proposition 5.2.7.4]). Let \(L\colon C \to C\) be a functor equipped with a natural transformation \(\eta \colon \id _C \Rightarrow L\) such that both maps \[ \eta _{Lx} \colon Lx \to LLx \qquadtext { and } L\eta _x\colon Lx \to LLx \] are isomorphisms for all \(x \in C\). Then the functor \(L\colon C \to \im (L)\) is left adjoint to the inclusion \(\im (L) \hookrightarrow C\), with unit \(\eta \). In particular, \(L\colon C \to \im (L)\) is a Bousfield localization.