Implication Details

Assumptions: natural numbers object, pointed

Conclusions: trivial

Reason: Let $(N,z,s)$ be a natural numbers object in a category with a zero object, denoted $0$ . The morphism $z : 0 \to N$ must be zero. The universal property applied to $A=1$ implies that $s : N \to N$ is an initial object in the category of endomorphisms. This exists, it is given by the identity $0 \to 0$ . Therefore, $N = 0$ . The general universal property now becomes: For all $f : A \to X$ , $g : X \to X$ there is a unique $\Phi : A \to X$ such that $\Phi(a) = f(a)$ and $\Phi(a)=g(\Phi(a))$ . Apply this to $g = 0$ to conclude $f = 0$ .