= Glued ring categories counterexample to regularity
Take two copies of the <category of rings>, identify their terminal zero rings, and adjoin a <strict initial object> $\bot$. Same-copy finite <categorical limits> are ordinary ring limits; products of nonzero objects in different copies are $\bot$. Ring surjections and $1_\bot$ are precisely the <strong epimorphisms>. They are stable under pullback along <monomorphisms>, so <image factorizations> exist and <Frobenius reciprocity for subobjects> holds. But pulling $\mathbb Z$ in one copy $\to0$ back along $\mathbb Z[x]$ in the other copy $\to0$ gives $\bot\to\mathbb Z[x]$. This is not epic: evaluations $x\mapsto0,1$ agree after precomposing with it. Thus the category is not regular, since strong epimorphisms would then be regular and pullback-stable.
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