Implication Details

Assumptions: natural numbers object, strict terminal object

Conclusions: one-way

Reason: By assumption, $z : 1 \to N$ is an isomorphism. Therefore, the terminal object $1$ is a NNO with $z = \mathrm{id}_1$ and $s = \mathrm{id}_1$ . This precisely means that for all $f : A \to X$ and $g : X \to X$ there is a unique $\Phi : A \to X$ with $\Phi = f$ and $\Phi = g \circ \Phi$ . In other words, we have $f = g \circ f$ , and therefore $g = \mathrm{id}_X$ (take $f = \mathrm{id}_X$ ), which proves the claim. (From here one can further deduce that the category is thin.)