A Finitary Adjoint Functor Theorem

Ranked locally finitely presentable categories are introduced, examples include categories of sets, vector spaces, posets, presheaves and boolean algebras. A finitary functor between ranked locally finitely presentable categories is proved to be a right adjoint if and only if it preserves countable limits. For endofunctors on vector spaces or pointed sets even countable products are sufficient. Surprisingly, for set functors there is a single exception of a (trivial) finitary functor preserving countable products but not countable limits.