  Let $F:\mathcal{A}\to \mathcal{B}$ be a left adjoint between abelian
categories and let $Ch(F)$ be the induced left adjoint on chain complexes. If
the abelian categories $\mathcal{A}$ and $\mathcal{B}$ are equipped with
sufficiently nice cotorsion pairs, then we find model structures on their
categories of chain complexes. This paper gives novel criteria under which the
induced adjunction becomes a Quillen adjunction. In particular, our criteria
can be checked on the level of the underlying abelian categories instead of
chain complexes. We do the same for adjunctions of $n$ variables and we also
show how our results imply that the flat model structure can be used to compute
the derived tensor product.
