The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of *Higher topos theory*, i.e. its inclusion functor has a left-exact left adjoint. Can this left adjoint ever have a further left adjoint, so that the hypercomplete objects would be an [essential localization](http://ncatlab.org/nlab/show/level+of+a+topos) or "level"?



Can hypercompletion be an essential localization?
Jacob's [answer](https://mathoverflow.net/a/213135/49) to my last question actually answers this one too: if the hypercomplete objects are coreflective as well as reflective, and the coreflector is *the same functor* as the reflector, then of course the reflector also has a left adjoint, namely the inclusion functor.


