Was Cod cold playable on ps5?

I would like to buy the new Cod Cold War today, I have decided on the Cross gen Edition, which means that I can still play it on the PlayStation 4 and when I switch to the PlayStation 5 I will download it again can without paying anything? In other words, I log into the PlayStation 5 with my main acc where the Cod cold was.

Correctly identified. You simply use the same psn account on ps5 as on ps4 and off you go

You can use the same psn account, so you can download all the games again because you have already bought them.

Thanks for the helpful answer.