Let $f:X\to Y$ be a morphism in a category $\mathcal{C}$. 
Let $m:I\hookrightarrow Y$ be the [regular image](https://ncatlab.org/nlab/show/regular+image) of $f$. This means that $f$ can be written as $f=m\circ e$, with $m$ [regular mono](https://ncatlab.org/nlab/show/regular+monomorphism) (i.e. being the equalizer of some pair of maps), and that moreover that it is universal, in the sense that if $f=m'\circ e'$ for some (other) $m':I'\hookrightarrow Y$ regular mono, then there exists $k:I\to I'$ such that $m=m'\circ k$.


Now here is the question: is the map $e:X\to I$ an epimorphism in general? I'm looking for either a proof, or a counterexample, or conditions under which it is true. For example, if $\mathcal{C}$ has equalizers and all monomorphisms are regular mono, such as in the category of sets, then $e$ is even an [extremal epi](https://ncatlab.org/nlab/show/extremal+epimorphism) (which in this case implies epi).


A reference would also be welcome.



Is the projection onto the regular image an epimorphism?
In general this is not true. The following counterexample appears in MacDonald-Stone, [The tower and regular decomposition](http://www.numdam.org/item/CTGDC_1982__23_2_197_0/), following Prop 1.8: let $\mathbf{3}$ be the ordinal $(0\le 1\le 2)$ regarded as a category, let $G^0\mathbf{3}$ be the disjoint union of 6 copies of the ordinal $\mathbf{2}$, indexed by the arrows of $\mathbf{3}$, and let $\epsilon:G^0 \mathbf{3}\to\mathbf{3}$ send each copy of $\mathbf{2}$ to the corresponding arrow in $\mathbf{3}$. Then the regular coimage of $\epsilon$ in $\rm Cat$ is a category $G^1\mathbf{3}$ that is like $\mathbf{3}$ but in which the morphism $0\to 2$ is not the composite $0\to 1\to 2$ (hence there are two morphisms $0\rightrightarrows 2$). In particular, the morphism $G^1\mathbf{3}\to \mathbf{3}$ is not monic. Thus $\rm Cat^{op}$ is a counterexample to your question.


Of course, a necessary and sufficient condition for this to be true in a category $\cal C$ is that every morphism of $\cal C$ factors as an epimorphism followed by a regular mono. If $\cal C$ has finite limits and colimits, then a sufficient condition for this to be true is that regular monos are closed under pushout (in which case $\cal C$ is [coregular](https://ncatlab.org/nlab/show/regular+category)).


