The act of making a judgment which is considered formal is to adjudicate. In this sense, adjudicate is a verb that represents the action of deciding a case. A judge or a mediator can adjudicate over a case depending on what has been agreed between the involved parties.
When the judge adjudicates on a case, the words they used are quoted verbatim and held as law in that instance. Both parties are expected to adhere to the decision of the judge. The term means basically to pronounce or declare.