We study when a numeration of the set of morphisms from a numeration to the other is well-behaved. We call well-behaved numerations ``acceptable numerations''. We characterize acceptable numerations by two axioms and show that acceptable numerations are recursively isomorphic to each other. We also show that for each acceptable numeration a fixed point theorem holds. Relation between Cartesian closedness and S-m-n property is discussed in terms of acceptable numerations. As an example of acceptable numerations, we study directed indexings of effective domains.
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