In legal terminology, to adjudicate refers to:

To adjudicate means to make an official judgment or decision about a legal matter. This process involves evaluating the evidence presented, considering legal arguments, and ultimately deciding the outcome of a dispute or issue at hand. The term is primarily used in the context of courts and legal proceedings, where a judge or tribunal adjudicates cases to reach a resolution based on the law.

The other choices do not accurately reflect the meaning of adjudicate. Postponing a ruling refers to delaying a decision, which does not involve the act of making a judgment. Administering punishment relates to enforcing consequences rather than deciding the outcome of a case. Transferring cases involves changing the jurisdiction or court in which a case is heard, which is separate from the act of making a judgment on a legal matter. Thus, the correct choice captures the essence of adjudicating as it pertains to the legal process.