Introducing a new cat into a household with already existing cats

From the American Association of Feline Practitioners

When introducing a new cat, initially isolate it in a separate room with its own food, water, litter box, and toys. This allows each cat to gradually adjust to the scent and sounds of the other cat. Once a veterinarian has deemed cats healthy, limited interaction
may occur under the door.

Continue the gradual introduction by exchanging bedding between cats, rubbing a cloth around one cat’s mouth and leaving that cloth in the other cat’s space, or rubbing a cloth alternately on each cat. As cats start to exhibit curiosity about each other, reward
friendly behavior with treats and praise.

At this point, short, supervised, direct interactions can begin. For the initial exposure, have one cat in a carrier and allow the other cat to approach. Feeding both cats at the same time can reduce stress as well. Following this, allow cats to walk around and sniff each other. Continue to reward friendly behavior. Gradually increase the supervised time together.

Do not leave the cats together unsupervised until several supervised interactions without aggression have occurred. The process of introducing a new cat may take several weeks. Older cats may need a quiet space away from kittens for an extended period of time. Friendly, well-socialized cats may adapt to each other rapidly.

Pheromonal products may ease the introduction but should be used in conjunction with a gradual introduction.