We've lived next to our neighbours for several years and are on friendly terms with them. They have an elderly cat who they keep outside. They used to call it inside at night, but this summer the cat was kept out the house (they put the food outside the door). Apparently this was due to them not being able to get rid of fleas. The cat has always been quite skittish and scruffy looking and just wants to sleep all day, curled up in flowerpot by the house. In the last year a couple of other cats have been coming into our gardens day and night and fighting the cat, it now has scars on ears and face.
Over the summer we've had our back door open and the cat has slowly grown more friendly and comes in to sleep in our conservatory. It now runs to us whenever we are around and comes into our house and accepts lots of petting and love. But it's getting colder and I don't want to shut my door and leave the cat outside all day. I've spoken to the neighbours and let them know of the cat's friendliness and sleeping in our conservatory. They say they don't mind at all and when I mentioned winter they said they will be bringing the cat in at night when it gets colder. Neighbour has joked that they'd get rid of the cat but their child doesn't want to. They pay hardly any attention to the cat apart from feeding it.
Basically I want to adopt the cat and let it stay inside my house day and night as it clearly wants to do. Would I be unreasonable to just straight out ask if we can adopt the cat?