So because he's willing to work through problems with his girlfriend, instead of just dumping her at the first sign of trouble, he's a... doormat?
It's more that he just let her continue until she decided that she didn't like the other guy with little to no objections. That hardly qualifies as "working through problems". Not to mention, cheating is hardly a mere "sign of trouble" by any stretch of the imagination.
Actually, I did object. However, considering she was never faithful to him for even a second, it wasn't exactly the worst thing in the world. And, she was honest with me the whole time. Additionally, I'll be honest, I do go very easy on her due to her issues (I've mentioned before, she's a paranoid schizophrenic, and has only in the last month gotten good about taking her meds, because she finally realized what would happen over time if she doesn't), but before this we had broken up several times beforehand. Since this happened, we haven't had any issues. If anything, it was what she needed to make sure she was right about being able to trust me and not be abandoned.