No expert, but I can tell you how the process went for us.
I brought a working line puppy home to a 7 year old cat that we rescued a couple years before that. We didn't allow Kona (the pup) to harass the cat in anyway, kept her on a leash, dragging it around the house, for several months so we could intervene. Our cat pretty much ghosted the first couple of weeks anyway, only came out when Kona was crated or sleeping. Eventually though, she got more comfortable and came out more often. Pup did go through a stage around 5 or 6 months old where all she wanted to do was chase the cat. UGH. I had a few moments during that time wondering if they'd ever be able to co-exist. During this time we upped the emphasis and training on the 'leave it' command. And she always, ALWAYS, was told to leave it when it came to the cat. If she even looked at the cat and got that look in her eye like she was about to give chase, "Leave it." Treat or praise.
So we did our part best we knew how. Our cat took it from there. A couple times when Kona invaded her personal space before we could make her back off, cat gave her a bat or two with a paw and a hiss. Giving her the cat version of a warning bark to back off. Kona has generally learned to respect her space, although now and again she needs a reminder. Our cat was an adult and had come from a home with multiple dogs before we adopted her. It became pretty obvious she knew her way around a dog. Kona has never shown any aggression toward the cat, just the chase drive.
Our cat has many places to hide if she wants to be away, including access through a cat door to the basement where Kona can't follow.
Kona is almost a year now and they are not best friends, but they co-exist perfectly fine. This morning Kona was laying on the bathroom floor next to me when I was getting ready for work and my cat was laying just above her on a ledge, not even looking at one another. They will even occasionally team up. We have caught our cat a few times batting food off the table or counter top on the floor to Kona. LOL