  For a locally presentable abelian category $\mathsf B$ with a projective
generator, we construct the projective derived and contraderived model
structures on the category of complexes, proving in particular the existence of
enough homotopy projective complexes of projective objects. We also show that
the derived category $\mathsf D(\mathsf B)$ is generated, as a triangulated
category with coproducts, by the projective generator of $\mathsf B$. For a
Grothendieck abelian category $\mathsf A$, we construct the injective derived
and coderived model structures on complexes. Assuming Vopenka's principle, we
prove that the derived category $\mathsf D(\mathsf A)$ is generated, as a
triangulated category with products, by the injective cogenerator of $\mathsf
A$. More generally, we define the notion of an exact category with an object
size function and prove that the derived category of any such exact category
with exact $\kappa$-directed colimits of chains of admissible monomorphisms has
Hom sets. In particular, the derived category of any locally presentable
abelian category has Hom sets.
