Basically I wonder, whether I "should" include the property of being epi in the definition of extremal epis / strong epis (/...) (dually for extremal monos etc.).
One hand it is terminology-wise a little bit strange if extremal epis may be non-epi for example. On the other hand, it is in accordance with the mathematical red-herring principle (semigroups need not be groups, skew fields need not be fields, etc.) and it seems more natural to keep the definitions simpler. Furthermore, in the presence of enough limits epicness follows. So I am inclined to to not include the epicness. But then again, I could just instead introduce "extremal pseudo-epis" or something similar too.
What would be considered common or "okay practice" here?