Implication Details

Claim: If a category has a parametrized natural numbers object and has a strict terminal object, then it is one-way.

Proof: Let $(N,z,s)$ be a parametrized natural numbers object. By assumption, $z : 1 \to N$ is an isomorphism. Hence, $(1,\id_1,\id_1)$ is a parametrized natural numbers object. The claim now follows from Lemma 3 here.