I'm reviewing some notes of mine on regular categories to try and get a better feel for regular, strong, and extremal epimorphisms, and this leads me to ask: is a category $\mathsf C$ regular if and only if $(\mathrm{ExtEpi},\mathrm{Mono})$ is a strong/orthogonal factorization system?

If anyone cares, here's my motivation.

In the category of sets, an image factorization of a function $f:A\to B$ is a composite $A\twoheadrightarrow \operatorname{Im}f\rightarrowtail B$ where $\operatorname{Im}f$ is in bijection with the set of fibers of $f$, the left arrow collapses fibers, and the right arrow takes a fiber to the value of $f$ on it. Thus image factorizations can be constructed functorially by taking for each function the coequalizer of its kernel pair - this collapses the fibers. Travelling further down this path leads to regular epis and the classical approach to regular categories.

Instead, we can define the image of an arrow $f$ (when it exists) to be the smallest subobject of its codomain through which $f$ factors. Often enough the first arrow in the factorization is an epi, so we define an *extremal epi* to be an *arrow* which doesn't factor through any proper subobjects of its codomain. Using this as a primitive notion, we can almost *define* the image of an arrow $f$ to be a subobject of its codomain which factors $f$ through an extremal epi. The only obstruction is the fact extremal epis are in general red herrings - they may not be epimorphisms. However, if $\mathsf C$ has equalizers, they are, and all is well.

Thus, in order for a category with equalizers $\mathsf C$ to have "image factorizations", it seems all we want is for extremal epimorphisms and monomorphisms to constitute a strong factorization system. The nlab writes:

A regular category is a finitely complete category which admits a good notion of image factorization

so tentatively I'm guessing that regular categories are precisely the finitely complete categories in which $(\mathrm{ExtEpi},\mathrm{Mono})$ is a strong/orthogonal factorization system. Is this correct?

**Added.** Just spilling some more thoughts. In general, if $\mathsf C$ admits intersections and pullbacks of monomorphisms, the prefactorization system $(\mathrm{StrEpi},\mathrm{Mono})$ is a factorization system. Moreover, in complete generality, strong epi implies extremal epi. Thus, if $(\mathrm{StrEpi},\mathrm{Mono})$ is already a strong factorization system, asking for $(\mathrm{StrEpi},\mathrm{Mono})$ to also be a strong factorization system amounts to asking every extremal epi to also be strong. What I'm having trouble seeing is how the notion of regular epi pops out here...