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You Can read about some of the possible pitfalls of not taking this or similar advice here
When you choose to have more than one cat, your chances for successful cat-to-cat relationships and a tranquil loving relationship with their owner are better if the two or three cats you choose are littermates. They do not need to be genetic sisters or brothers - just to be or have been kittens that grew up and nursed together. If not that, at least cats from the same colony source that show the traits of friendship in their day-to-day setting.
When you are observing cats in a shelter environment or homeless cats in the field, observe them from a distance. When you see two or three mature cats that consistently stay close to each other, perhaps lying together in physical contact one using the other as a pillow or licking each other (allogrooming), those cats are almost certain to make good housemates. (ref)
The mission of animal shelters is to place as many of their wards in homes as possible. Several I have worked with have a no-return policy. If it is an option, find a shelter that will let you foster two or three kittens or cats for one to two months with the understanding that you can return them, no questions asked, no assigned guilt, if their relationships and behavior in your home is not what you seek. I would delay naming those kitties for that period, once you have given them names, you will find it exceedingly difficult to give any of them up.
Many cat observers divide feline personalities into three general types. But the divisions in temperament between the three are not nearly as crisp as I list them below - no more than human personality types are crisply divided. However, once the general personality of a cat is established, it rarely if ever changes.
1) My favorite cat is the sociable, confident and easygoing feline. It will come up to you and meow or chirp even when you are a total stranger - like the one in the top image. When it does, the cat will carry its tail almost straight up - with a slight curl at the tip. After circling you, it will rub on your leg and purr. If you extend your hand, it will sniff it. It might roll over back and forth at your feet. It is curious and will chase a toy, a feather or a string if you offer it one. It passes by you again and again to be stroked. As it does, it makes trill and chirping sounds. If you pull gently on its tail, it will simply turn around. If it is in an animal shelter cage, it is lounging near the front. It might yawn in your presence. These cats make excellent pets. They have loving relationships with their owners and usually get on well with their feline housemates.
2) The second type of cat is shy, suspicious and fearful. It rarely blinks. It will withdraw from you if you approach it. It resents handling. When forced into situations that are too intimate, it is likely to claw or bite in an attempt to escape. It is also likely to take a swipe at you if you gently pull on its tail. If it is in an animal shelter cage, it is in a rear corner. It is too tense to yawn in your presence. These cats were never properly socialized as kittens. They carry a lot of emotional baggage. When adopted, they are over-represented in cats with inter-cat and inter-owner issues.
3) The third type of cat is excitable, nervous and tense. It ignores or is hostile other cats. It regards you with an intense stare. It stands its ground when approached and will spit, hiss or growl. Its only motivation to approach its caretaker is to be fed. These are the cats most likely to attack their owners and housemates. With time, they may learn to tolerate you; but they will never be affectionate to you.
Animal shelters are places of turmoil for cats. Bright colors, wall prints and spotless stainless steel cages make animal shelters very presentable to visitors. But they do not impress cats. Even a very sociable cat is likely to be fearful initially in such an environment. The caretakers of these cats are in a conundrum. If they wait long enough to truly judge the temperament of the cats and kittens that come in, they cannot keep up with the inflow of new cats. If they maintain high numbers of cats under evaluation, they exhaust their financial resources and space. If they maintain incoming cats and cats emotionally chilling down in the same facilities, they experience periodic outbreaks of disease. (ref) If they keep cats long enough for them to revert to their best dispositions, other health issues can be discovered that require expensive veterinary care. Contrary to what most people suppose, your local shelter organizations receive no financial support from groups like the ASPCA or HSUS - at least none of the many I have ever volunteered at did. (ref) Not only the cats are stressed by the realities of this situation; it is highly stressful to the dedicated people providing their care. (ref) Over the years, I have seen the emotional exhaustion, turnover and negative effects on the general health of these good folks who are attempt to assist the animals. (ref1, ref2) It is not that different in other high-stress, emotion-laden jobs that take a toll on one's empathy. (ref) Is it any wonder that shelter staff is motivated to places as many cats in homes as they can, just as fast as they can? For very explainable reasons, what happens down the road is not their primary focus. But it needs to be your primary focus.
If your cat will come from a shelter, spend some time looking over their available cats using some of the methods I mentioned earlier to judge their underlying dispositions. Sign an agreement with the staff to adopt the cat that you choose. Although it is standard dogma to do so, I prefer those that do not insist that you spay or neuter kittens too young. That brings up a whole different set of later health issues. (ref)
Be sure the cat is already free of fleas, ticks, internal parasites, ringworm and ear mites. The cat needs to be confirmed feline leukemia and feline AIDS negative. I prefer that the cat’s health has already been checked by their participating veterinarian. There is nothing as disconcerting to a veterinarian as to having to convey bad news to happy clients who drop by with a newly adopted cat – be that anti-social or physical health issues. Have a person in authority at the shelter agree that you can return the cat to them, no questions asked, if things do not work out.
Later in the day, go to a store and buy a spacious, high-quality rigid plastic pet carrier. Not the cardboard ones the shelter will give you. Use it to pick up your new cat. While you're there, pick up a cat collar and have a tag with the cat's name and your phone number made on the spot. Don't buy a break away collar because you aren't going to let your cat run loose unattended anyway - are you? Let the shelter staff adjust the collar to a comfortable but snug fitting. Have the staff bring the cat to you and place it in the carrier. Do not let them take the carrier to the back where it can pick up the odors of other cats. Do it on a non-hectic morning for you and for their staff - after their feeding and cleaning chores are over. Do not bring your children to cat shelters unless they understand that you, not them, will choose the cat.
If you already have other cats at home, those that previously accepted newcomers are more likely to accept them again. The reverse is also true. Hopefully you have a room entirely separated (but not too distant) from the cats you already have. Preferably a room that is not a preferred location for one of your resident cats. It should be a quiet room in which you can place separate food, water, litter box, and enrichment activities and hiding area. A cardboard box or two will do. Some find that the liberal use of Feliway-type products in the new and resident cat’s quarters during this adjustment period is helpful.
I would maintain that situation for about a week – perhaps more if the resident cats or the newcomer appear tense or agitated, perhaps less if everyone remains mellow. After the first day or two, depending on how calm things appear, take two small pieces of your apparel or one of your used small bathroom towels . Mark them and stroke each cat down with one of them. Then place the item in a corner of the living areas of the opposite cats. If all goes well, repeat that round-robin from day to day until the cats show no interest in sniffing the cloth. I worry more about cats not getting along that show no interest in sniffing the scents on the cloth than those that do.
Gradually increase the contact between the cats. Perhaps there is enough space under the door to the new cat’s room for your resident cat(s) to explore the new arrival. If not, devise some way for the cats to see each other and observe what happens. Which one’s tail goes up, which goes flat, whose ears lay back, which - if any- tense up, spit, growl or hiss. Rubbing the divider or purring is a very positive sign for either cat. The one that doesn’t is likely to be the most problematic with the new relationship.
When you do attempt to allow the cats free access to one another be sure to have extra food stations, litter boxes, water dishes and hiding areas. The more the better. You will need to chaperone them for a while until you are certain that they have become compatible. Play with both cats for short periods.
If altercations occur, go back to the previous living situation, give them more time and then try again. Keep towels and heavy gloves handy to separate cats if necessary. Once they appear to tolerate each others close presence, let them develop their relationship at their own pace. Don’t attempt to push them closer to each other or compete for resources, snacks and toys in an attempt to break the ice. That does not work and is counter productive.
You can read another veterinarian’s thoughts about introducing cats to each other here.
Cats that are uncomfortable with each other rarely interact. When they do, it is often aggressively over possession of choice resources (food, toys, favorite resting area, etc.) One will often leave the room when the other arrives. They tend to watch each other intently, giving none of the indications of inter-cat affection I mentioned earlier. They are tense when circumstances require that they be in close contact. They never sleep near each other nor do they groom each other. One or both may loose their toilet training. An occasional blink is a sign of a relaxed, contented cat. One rarely sees that in a cat uncomfortable in its living arrangements.