Separating tower site with vanishing product ($A_n\twoheadrightarrow 1,{\prod }_n{A}_n=0$)

Consider a site indexed by natural-number objects with arrows only downwards, the cancellation condition $fh=gk\Rightarrow f=g$ for equally typed $f,g$, and the separation condition that arrows $f:m\to n$, $g:m\to n+1$ admit $h,k:m+1\to m$ with $fh=fk$ but $gh\ne gk$. Its dense-sieve sheaf topos is a Boolean topos and a two-valued topos. The representables are separated. No nonempty partial map $y_n\to y_{n+1}$ exists, by the separation condition and naturality. The associated sheaves $A_n$ consequently admit no maps $y_n\to A_{n+1}$. Each has nonzero support and maps epimorphically to one, but their product has an empty value at every stage $m$, because its $A_{m+1}$ factor does.