Every volunteer had a predesignated responsibility to ensure the event went well.

predesignated 是由前缀 pre-（表示“预先”）和动词 designate（指定）组合而成，再加上 -ed 构成形容词形式。pre- 表示“在……之前”，designate 意思是“指定、指派”。predesignated 合起来表示“预先指定的”。