  We study selfadjoint functors acting on categories of finite dimensional
modules over finite dimensional algebras with an emphasis on functors
satisfying some polynomial relations. Selfadjoint functors satisfying several
easy relations, in particular, idempotents and square roots of a sum of
identity functors, are classified. We also describe various natural
constructions for new actions using external direct sums, external tensor
products, Serre subcategories, quotients and centralizer subalgebras.
