  We prove that given a Grothendieck category G with a tilting object of finite
projective dimension, the induced triangle equivalence sends an injective
cogenerator of G to a big cotilting module. Moreover, every big cotilting
module can be constructed like that in an essentially unique way. We also prove
that the triangle equivalence is at the base of an equivalence of derivators,
which in turn is induced by a Quillen equivalence with respect to suitable
abelian model structures on the corresponding categories of complexes.
