Is that also Warzone included because when I search for Warzone in the playstation store it shows that I'm currently installing it, but there's nothing from Warzone in the main menu
Warzone is free and modern warfare costs 70 euro I think. If you bought it you got both
Did you only get warzone (free) or the full game?
The full game
Yes then you have everything Coop campaign and warzone
Just make sure that you download all the different data such as data pack etc., everything should be in your download list