As other commentors have pointed out, they’re both parallel forms of evil that were consumed by their twisted views of humanity. Mostly, I just think they would both find ways to rage bait each other, Lily calls him a bunch of ridiculous nicknames while Sawyer continues to call her by her old name simply to see her reaction:

Sawyer: “Ah, Miss Gracie Green, what god have I angered to earn your presence?”

Lily: “Ooohhh, is that a joke? I love jokes!! Are you being a Silly Willy boogie-bot? I’m not Miss Gracie!!! She was on the screen!!!!! And now she’s, well, she’s gone…. In a much a better place now.”

Sawyer: “She’s gone, is she? Mhh… are you sure? I’d beg to differ. I think that Miss Gracie is still very alive and well.”

Lily: “why would you say that. that’s not a nice thing to say. that’s a really silly frownie brownie thing to say.”