cats

Excellent question. There are some foundational skills the dog needs to be able to participate safely and effectively in the process. This process was also accelerated by the cat being a ho for human attention, making it easier to transfer some of that affection / prosocial behavior.

Cat train the puppy for safety

1. leave cat claws untrimmed for a month before getting the puppy and at least 6 months in
2. Dog was crate trained so unsupervised interaction was a non issue
3. Distract puppy every time her gaze locks onto the cat for more than a few seconds (message: cat = not interesting)
4. When dog learns belly, teach her to do it whenever the cat is <3ft or so away

Habituation

1. Use treats to slowly lure them closer to each other (note: dog must have a solid grasp of "leave it" / be able to understand when a treat is not hers
2. Alternate petting them including alternating which hands are petting them respectively during the same interaction. Specifically try to rub their respective scents on each other near their faces / snouts. The cats scent is best obtained from the cheekbones, while the dog's scent is usually easy to obtain anywhere on them. I did find the dog liked to lick my hands, so I would rub her saliva on the cats cheeks (it helps to know and leverage exactly what your cat considers a "high value petting.")
3. Group cuddles: what you see here. Everybody keeps getting treats as long as you're willing to sit in the cuddle pile. Never ignore or discourage spontaneous attention seeking behavior from either animal in this context.