It has been so long since I adopted a cat. They usually appear spontaneously and move in.
A few years ago, we moved into a neighborhood with a very prolific feral cat. Eventually, we trapped her—she was too smart to be trapped, but her kitten helped—and had her fixed. Her last litter of kittens was about 4 months old. We took them in and tamed them. The two girls remain skittish, while the boy is the most friendly little guy. These “kittens” are now 9 years old.
My cats range from 7 to 11. They’re still my kittens.