on the understanding that
Dependent on agreement to a specific named condition. I'm willing to lend you the money, on the understanding that it will be repaid with interest. His parents agreed to pay for his college tuition, on the understanding that he would pursue a career in medicine.
on the underˈstanding that...
(formal) used to introduce a condition that must be agreed before something else can happen: They agreed to the changes on the understanding that they would be introduced gradually.