We show that a complete abelian category with an injective
cogenerator has exact direct limits (i.e. satisfies Grothendieck's
AB5 condition) if and only if one (or any) injective cogenerator
is pure-injective in the sense of Jensen and Lenzing (i.e. the
summing maps from coproducts extend to products). The result is
motivated by tilting theory and the proof uses a representation
theorem for the dual setting of cocomplete abelian categories with
a projective generator via additive monads on the category of
sets. This is an account on joint work with Leonid
Positselski (arXiv:1805.05156).