Dear Jean,
What would you call a category X such that the functor X --> 1 is full and faithful? Please don't tell me what they are, I know that.
Sticking to the pattern I suggested I'd call it "essentially subterminal".
Non evil is essentially evil. I rather like this conclusion, don't you?
Of course, that's brilliant dialectics! I'd expect the people abhoring evilness would say that full and faithful and essentially surjective is an "evil" notion of equivalence as opposed to the "good" one of adjoint pair where unit and counit are isos. The latter makes sense in any 2-category whereas the former doesn't. However, often you just get the "evil" version when not having a strong form of AC (for classes) available. That's why your dialectics definitely applies! Best regards, Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]