The study of inductive and coinductive types (like finite lists and streams,respectively) is usually conducted within the framework of category theory, whichto all intents and purposes is a theory of sets and functions between sets.Allegory theory, an extension of category theory due to Freyd, is better suited to modelling relations between sets as opposed to functions between sets.The question thus arises of how to extend the standard categorical results on theexistence of final objects in categories (for example, coalgebras and products)to their existence in allegories. The motivation is to streamline current work ongeneric programming, in which the use of a relational theory rather than afunctional theory has proved to be desirable.In this paper, we define the notion of a relational final dialgebra and prove, for animportant class of dialgebras, that a relational final dialgebra exists in an allegory if and only if a final dialgebra exists in the underlying category of maps. Instancessubsumed by the class we consider include coalgebras and products. An important lemma expresses bisimulations in allegorical terms and proves this equivalent to Aczeland Mendler's categorical definition.