Well, my full bred oldest came home and met the cat, all was fine the puppy outgrew the kitty..an Kitty was dragged across the floor like a toy. Cat by itself moved to the basement and refused to come back upstairs. Original CAT moved to Grandma's House eventually.
Since that time a stray cat adopted us...as and older dog our girl herds but guards her pet kitty. the new Kitty smaller than the first is fine with living conditions,but then she chose us.
It all depends, The cat may rule the house in your case.
Best Wishes for you!