Fit-to-fly certificate
A vet's note, usually issued within days of departure, stating the animal is healthy enough to fly. Some airlines and routes ask for it on top of the border paperwork: it is an airline requirement, not a border one.
It is worth separating two things that look alike. Border paperwork is what a country demands to let the animal in. A fit-to-fly note is what an airline may demand to let it board, and it is usually a recent vet certificate stating the dog is healthy enough for the flight. It is an airline requirement, not a border one: border paperwork does not replace it, and having it does not exempt you from that paperwork. If your airline asks for one, check how many days before departure it accepts.