Pre-recursive categories

How can we obtain in a natural way the primitive recursive functions in categories? In this paper, we study the free ‘cartesian closed category with a natural numbers object (in the sense of the Peano-Lawvere axiom)’ generated by the empty category. In this category, every morphism 1 → N represents a natural number and every morphism N → N represents a function. Furthermore, the set of functions represented by the morphisms of this category contains strictly the set of primitive recursive functions and is strictly contained in the set of recursive functions. Then, we see that this category is a categorical version of Grzegorczyk's recursive functionals of finite type, with the addition of product types.