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If they don’t already know you have a cat, let them know that you do, that you are training him to go outside, and to stay within the boundaries of your yard.  Explain that he is not supposed to leave the yard, and that if he comes into their yard, they can either shoo him back to your yard, or they can call you.  Ask them to shoo your cat firmly, but without trying to frighten the cat. If they frighten him too much, he may run into the road, or somewhere other than your yard. If you have a fence already, you can skip this step. If you do not, you should use something, such as small flags or large stones, to mark the boundaries of your yard. It is best if you get a children’s squirt gun that can shoot very far. This way, you can hide if you need to spray your cat. It is important that your cat does not see that the water is coming from you. If the does, he will wait until you are not around to do whatever they want to do.  An exception to this is if you see that your cat needs to immediately be redirected because of danger (e.g. if you see he is headed towards the road, and there is an oncoming car). In dangerous situations, it is more important that you redirect him quickly, regardless of whether he knows you are doing it.  You should also try to avoid spraying him in the face, unless it is to protect him from an immediate threat.. In his first days outside, you will want to choose a quiet time of the day, right before your cat usually takes a nap. Call your cat by saying “Here!” (with his name) loud enough so that he can hear you. When he comes to you, praise him while you put on his harness and leash.  In the beginning, you will carry your cat outdoors to demonstrate that you are in charge of the situation, but this will change later. You should set him down just a few feet outside the door so that he is facing the house.  Pet him immediately, so that he gets used to being touched while outdoors. While holding on to his leash, allow him to sniff around the yard freely. Make sure that you have a firm grasp on the leash in case anything frightens him. You don’t want the leash to slip out of your hand if this happens.  Every few minutes, you need to pick him up and pet him while also saying his name. Once you have done this, set him down again facing the direction of your home.  It is important that you do this several times before taking him back inside. If he learns that it means he has to go back inside when you pick him up, he will try to run and hide. Not only will this be unpleasant for your cat, it also increases the chance that he will escape from his harness. Cats are masters of escaping. If you pull his harness tight using the leash, it will be much easier for him to wriggle out of it, and once he figures this out, he will do it every time.  If your cat jerks away from you because he is frightened, pick him up as soon as possible (step on the leash if necessary). If you don’t grab him, there is a good chance that he will get out of his harness. If something does frighten him, pick him up and take him indoors immediately. Especially if he doesn’t calm down quickly. Give him a treat, and try again the next day. While it is generally not a good idea to take him inside as soon as you pick him up, a situation where he wants to run away warrants this. You want to discourage him from bolting out of the yard. Any time your cat walks towards you or the house, you should praise him. Keep a toy, such as a stick with a string and toy mouse attached to it, on the porch. Try to engage him in playing with the mouse close to the house. During the first three days, the goal is to keep him near the house, and to avoid situations where you have to tell your cat “no.” If he is on a leash, you will have more opportunity to guide him towards the home (without dragging him!).  Start with session of 5-10 minutes. You can use positive things such as toys and treats to keep him interested in staying near the house. As soon as you need to bring him inside, pick him up, pet and praise him for a minute or two, then take him inside. Once you are inside, reward him with cuddling (if he enjoys it) and a treat. Tell him he did good. After three successful days, you can take him outside, and let his leash drag on the ground. Continue to reward him for staying close to the house, but also allow him to explore a bit.  The point here is to teach your cat that as long as he stays inside the boundaries, he gets to roam around, but once he crosses those boundaries, play time will be over.  Just as in the first three days, you should continue to pick him up every few minutes, pet him, and praise him. The point here is to remind him that you aren’t necessarily a signal that he has to go back inside. Whenever your cat approaches the boundary of the yard, stay close to him. As soon as he steps on or over the boundary, say “no.” You should avoid using his name, and you should not yell it so loudly that it will frighten him.  Instead, say “No,” and get between him and the outside boundary. Gently shoo him back in the boundary by using hand motions, or by gently pushing him back into the yard.  Never hit your cat. Hitting an animal only damages their trust in you. Any time you see him approaching the boundary, you should do this. You can say “No” or “Uh-uh.” If he crosses the boundary anyway, go and put him back inside the boundary. If he crosses the boundary a third time, then you should take him inside. Remember though, that you should still reward him when he comes indoors. He needs to learn that going inside is a good thing. A good strategy is to plan his outdoor time right before his normal meal time so that he has something to look forward to. Otherwise, you can try luring him in with toys and/or treats. If he ignores you, and stays outside, then go over to him and pick him up. Walk around with him for a few seconds while petting him. Apologize, and say that you have to go inside now. Once inside, praise him, and give him a reward. Once he goes outside without his leash and harness, remain cautious. You should still keep a constant eye on him. If he steps on or over the boundary, give him a verbal warning immediately. If he violates that warning even once, take him back inside.  At this point, you should also try to hide for a few seconds at a time, or at least stay out of his line of sight. The cat needs to understand that the rules apply whether you are there or not. You can also hide with your water gun. If he steps on or over the boundary, spray him with the water. Again, do your best to avoid being seen. Over the span of a few months, your cat should come to understand that the yard is where he is allowed to be, and that going out of the yard is unacceptable. Once you feel comfortable that he will not leave the yard, you can leave him alone, but do not forget to check on him—frequently at first, and then occasionally. You may also need to have reminder training sessions where you stay out with him the whole time, picking him up occasionally to keep him comfortable with being touched while outside.

Summary:
Let your neighbors know about your cat. Mark the cat’s boundaries. Have a spray bottle handy. Set aside three days for initial training. Take your cat outside on a leash. Allow him to walk around. Avoid dragging your cat by the leash. Reward your cat for coming towards you or the house. Keep the first sessions short. Work up to letting the cat roam more freely. Watch closely. Encourage him to come inside on his own. Be persistent.