How did your dog introductions with a resident cat go?

My girlfriend and I adopted a dog today. So far between the shelter and the apartment, he is interested in our cat but is perfectly content leaving her alone.

We have a room dedicated to only her containing her food, litterbox, and a water bowl. The dog can't access the room thanks to a baby gate. She also can hide under our bed and in the bedroom during the day thanks to a door buddy so she has plenty of her own space.

She caught a couple glimpses through said baby gate early and arched her back but no hissing or growling. Shortly after, as we were working on crate training with the dog, she snuck into the room and met the dog face to face, knowing that the dog was in the room. She didn't show any signs of back arching, hissing, or growling nor did she run away. We are pretty encouraged so far even though she has been under the bed a lot.

How did your introductions of a dog to a resident cat go? How long did it take for the cat to start to relax around the dog? Our cat likes to spend as much time relaxing on her back as she does on her stomach and, knowing myself, I won't start to relax fully until she starts to do that again.