It is unlikely you will be able to avoid action between the Pets under the title “Who’s boss”. The struggle will be fought for any place in the house. Even the seat on your lap. So be prepared for the fact that, while your Pets will get along, peace in your home will not.
And if you already have an adult cat or dog, you should not immediately tune in to the fact that he disliked the new. It is possible that your pet will treat the new pet as a kitten or a puppy: will take care of it in every possible way and play with it.
If the animals in varying degrees, show aggression, zeal to the owners, hostility to other animals, it is good to think before you have a second pet, because you can jeopardize health and life of the new pet.