conatural numbers object

A conatural numbers object in a category with an initial object is a natural numbers object in the opposite category. This is not standard terminology and should not be confused with the set of conatural numbers studied in computer science, which dualizes natural numbers in a different way. Explicitly, if $0$ denotes the initial object, a conatural numbers object is a triple $$(N,\, z : N \to 0,\, s : N \to N)$$ satisfying the following universal property: for all $a : X \to 0$, $g : X \to X$ there is a unique $\Phi : X \to N$ such that $z \circ \Phi = a$ and $s \circ \Phi = \Phi \circ g$.By the proof of this result, if countable powers exist, the object $\prod_{n \in \IN} 0$ carries the structure of a conatural numbers object, which simplifies to $0$ in many categories. This is why this notion is not very interesting.