  We show that Quillen's small object argument works for exact categories under
very mild conditions. This has immediate applications to cotorsion pairs and
their relation to the existence of certain triangulated adjoint functors and
model structures. In particular, the interplay of different exact structures on
the category of complexes of quasi-coherent sheaves leads to a streamlined and
generalized version of recent results obtained by Estrada, Gillespie, Guil
Asensio, Hovey, J{\o}rgensen, Neeman, Murfet, Prest, Trlifaj and possibly
others.
