This comment says that in the category of a partially ordered set, every arrow is an epi but no non-identity arrow has a right inverse.
My understanding is that the category in question is one where there is at most one arrow between any two objects, and there is an arrow $a\to b$ iff $a\le b$.
I'm having a trouble verifying the above claim (at least its second part).
For the first claim. Consider an arrow $f:a\to b$. To show it's an epi, we need to show that $hf=h'f\implies h=h'$ for all arrows $h,h':b\to c$. Well, suppose $h,h':b\to c$ are arrows such that $hf=h'f$. Since there is at most one arrow between $b$ and $c$, $h=h'$. This completes the proof. The assumption $hf=h'f$ is not even needed, right?
For the second claim. Suppose $f:a\to b$ is a non-identity arrow (this means $a\le b$). Assume it has a right inverse $r:b\to a$ (i.e., $b\le a$) for which $fr=1_b$ holds. What does it contradict to? I don't quite understand what statement $fr=1_b$ means in the language of $\le$.