A strongly saturated class gives a localization

Let $$C$$ be a presentable category and let $\Sigma$ be a strongly saturated class of small generation.

The inclusion $\Loc_{\Sigma}(C) \hookrightarrow C$ of the $\Sigma$ -local objects in $$C$$ admits an accessible left adjoint $L\colon C \to \Loc_{\Sigma}(C)$ .
The category $\Loc_{\Sigma}(C)$ is presentable.
We have $\ker(L) = \Sigma$ , i.e. a morphism in $$C$$ is inverted by $$L$$ if and only if it lies in $\Sigma$ .
For every category $$E$$ , precomposition with $$L$$ induces a fully faithful functor
$\Fun(\Loc_{\Sigma}(C),E)\longrightarrow\Fun(C,E)$
whose essential image consists of the functors that invert every morphism in $\Sigma$ .

References

Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.