model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
A simplicial Quillen adjunction is an sSet-enriched Quillen adjunction: an enriched adjunction
of sSet-enriched functors between simplicial model categories $C$ and $D$, 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 $(\infty,1)$-derived functor under the model of (∞,1)-categories by simplicially enriched categories.
Let $C$ and $D$ be simplicial model categories and let
be an sSet-enriched adjunction whose underlying ordinary adjunction is a Quillen adjunction. Let $C^\circ$ and $D^\circ$ be the (∞,1)-categories presented by $C$ and $D$ (the Kan complex-enriched full sSet-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of adjoint (∞,1)-functors
On the decategorified level of the homotopoy categories these are the total left and right derived functors, respectively, of $L$ and $R$.
This is proposition 5.2.4.6 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).
If $C$ and $D$ are simplicial model categories and $D$ is a left proper model category, then for an sSet-enriched adjunction
to be a Quillen adjunction it is already sufficient that $L$ preserves cofibrations and $R$ 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 $D$ the left Bousfield localization of a simplicial left proper model category $E$ at a class $S$ of morphisms, for checking the Quillen adjunction property of $(L \dashv R)$ it is sufficient to check that $L$ preserves cofibrations, and that $R$ takes fibrant objects $c$ of $C$ to such fibrant objects of $E$ that have the property that for all $f \in S$ the derived hom-space map $\mathbb{R}Hom(f,R(c))$ is a weak equivalence.
simplicial Quillen adjunction
On the enhancement of plain Quillen adjunctions between left proper combinatorial model categories to simplicial Quillen adjunctions: