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http://media.comicvine.com/uploads/3/32493/1411880-16247_400x600_large.jpg

vs

http://media.comicvine.com/uploads/0/8632/599923-wonderwoman26_large.jpg

Has Shaggy displayed anything close to suggesting his soul/psyche could handle being ravaged by Genocide?

Shaggy Man has no soul or psyche. He's an artificial being.

I thought he was once a guy who was transformed into what he is now? Or is that The General? I thought the General was Shaggy Man, or at least turned into it or something.

I think the OP wants the original, mindless Shaggy Man going by the picture.