Lizzy must be more of a 'challenge' for someone like Lady Catherine than her usual circle provides. While she clearly does not warmly embrace Lizzy's personality and wit, I think she may be somewhat captivated by her. "Not so's you'd notice" of course---!---but I think Lizzy provides a little color and intellect for Lady C. that no one else in her life does. I have known a Lady C. type of person; almost the only people she 'begrudgingly respected' (and even looked forward to sparring with, I think) were those few who stood up to her in conversation (and behavior.) I suspect the Lady Catherines of the world are actually a bit lonely and bored with the cast of characters they can 'manage' and manipulate!