Does he ask to go out often?

I don't think he is depressed from missing outdoor life - both of my two former stray cats appeared to be very grateful to have a home, especially in a harsh weather, and they ate eagerly and stayed put once stomach was full during the first few months.  My vet told me that stray cats don't play and the newly intaken cats were still in a surviving mode.  Mostly their issues are over-eating and gaining too much weight in a short period of time.