Cardinality functions in allegories

In this paper we want to investigate three notions of the cardinality of relations in the context of allegories. Two of the three axiom systems are motivated on the existence of injective and surjective functions, respectively. Since those notions may differ we provide a third unifying definition of cardinality. In all cases we provide a canonical cardinality function and show that it is initial in the category of all cardinality functions over the given allegory.