We prove a generalization of Alex Heller's existence theorem for recursion categories; this generalization was suggested by work of Di Paola and Montagna on syntactic -recursion categories arising from consistent extensions of Peano Arithmetic, and by the examples of recursion categories of coalgebras. Let = ⟨⟩ be a uniformly generated isotypical -subcategory of an iteration category , where is an isotypical object of . We give calculations for the existence of a weak Turing morphism in the Turing completion Tur () of when is separated; i.e., when connected domains in are jointly epimorphic. Our proof generalizes as follows. Let be a separated iteration category and let : → be an iteration functor; i.e., a functor which preserves domains, coproducts, zero morphisms and the iteration operator; it is crucial for the generalization that an iteration functor need not preserve products. If is faithful, then Tur () is a recursion category
