COFUND: An entity with a successful COFUND project in Horizon 2020 would like to submit a new proposal in Horizon Europe. Is it true that if an organisation succeeded in a Postdoctoral Programme, it is advised to opt for a Doctoral Programme in case of a new submission? And vice versa?

There are no such restrictions. If the proposal is excellent, it will get funding, no matter if it is only a Doctoral Programme or a Fellowship Programme. There are examples of organisations with consecutive programmes.