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?

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    $\begingroup$ I'm not going to take a stand on your question, but I don't think the comparison with semigroups is a good one: the prefix "semi" clearly indicates a weakening, while adding an adjective like extremal (or especially strong!) suggests a strengthening. I agree that skew field is unfortunate terminology for this reason. $\endgroup$ – Alex Kruckman Apr 9 '16 at 21:04
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    $\begingroup$ Please don't introduce "extremal pseudo-epi"! Johnstone's Sketches of an Elephant defines a cover to be an extremal epimorphism that is not necessarily an epimorphism, which is an alternative terminology also mentioned on the nlab. Perhaps perversely you can use cocover for extremal monomorphisms that are not necessarily monomorphisms. $\endgroup$ – Vladimir Sotirov Apr 15 '16 at 3:04

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