related by the Dold-Kan correspondence
of sSet-enriched functors between simplicial model categories and , such that the underlying adjunction of ordinary functors is a Quillen adjunction between the model category structures underlying the simplicial model categories.
Simplicial Quillen adjunctions model pairs of adjoint (∞,1)-functors in a fairly immediate manner: their restriction to fibrant-cofibrant objects is the sSet-enriched functor that presents the -derived functor under the model of (∞,1)-categories by simplicially enriched categories.
Let and be simplicial model categories and let
be an sSet-enriched adjunction whose underlying ordinary adjunction is a Quillen adjunction. Let and be the (∞,1)-categories presented by and (the Kan complex-enriched full sSet-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of adjoint (∞,1)-functors
This is proposition 18.104.22.168 in HTT.
The following proposition states conditions under which a simplicial Quillen adjunction may be detected already from knowing of the right adjoint only that it preserves fibrant objects (instead of all fibrations).
to be a Quillen adjunction it is already sufficient that preserves cofibrations and just fibrant objects.
This appears as HTT, cor. A.3.7.2.
This is in particular useful for finding simplicial Quillen adjunctions into left Bousfield localizations of left proper model categories: the left Bousfield localization keeps the cofibrations unchanged and preserves left properness, and the fibrant objects in the Bousfield localized structure have a good characterization: they are the fibrant objects in the original model structure that are also local objects with respect to the set of morphisms at which one localizes.
Therefore for the left Bousfield localization of a simplicial left proper model category at a class of morphisms, for checking the Quillen adjunction property of it is sufficient to check that preserves cofibrations, and that takes fibrant objects of to such fibrant objects of that have the property that for all the derived hom-space map is a weak equivalence.