I've been trying to get my head around the various types of epimorphisms you get in category theory, but I can't see why anyone uses "extremal" epis as opposed to the slightly less general notion of "strong" epis.
Every strong epi is extremal; extremal epis can be proved strong if you have pullbacks; so the notions coincide in pretty much any category you're likely to be working in. So for instance in Top the extremal epis = strong epis = quotient maps (as opposed to any old surjective continuous map).
What's more, the definition of a "strong" epi arises naturally when you try to work out what conditions you need to put on an epi to get unique epi-monic factorisation. Try proving that Set has unique epi-monic factorisation, for instance; you'll end up proving a lemma that states all epis in Set are strong.
The definition of "extremal", in contrast, seems to come out of nowhere. So why bother with extremals at all? Is there any use or motivation for the definition, or is it just some kind of historical hangover?
Thanks in advance for any light you can shed on this.