= Shift epimorphism of the natural-number presheaf classifier
{title2=$F_n(k)=\max(n,k-1),\quad F_n(\infty)=\infty$}
In the covariant <functor category> $[\mathbb N,\mathbf{Set}]$, the classifier at $n$ consists of thresholds $k\geq n$ and the empty threshold $\infty$. Its transition to $n+1$ sends $k$ to $\max(k,n+1)$. The displayed maps commute with transitions and are all surjective, so define an <epimorphism>. They identify $n$ and $n+1$, so the endomorphism is not an <isomorphism>.
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