  We prove that the exactness of direct limits in an abelian category with
products and an injective cogenerator J is equivalent to a condition on J which
is well-known to characterize pure-injectivity in module categories, and we
describe an application of this result to the tilting theory. We derive our
result as a consequence of a more general characterization of when inverse
limits in the Eilenberg-Moore category of a monad on the category of sets
preserve regular epimorphisms.
