What is the name of the medieval church court system dealing with religious law?

The story behind the answer

Because only the Church’s own legal system claimed authority over spiritual matters, it handled cases on doctrine, sacraments, clergy discipline, marriage, oaths, wills, and heresy. These were tried in ecclesiastical courts and punished with spiritual penalties like penance or excommunication, unlike secular systems: common law governed English royal courts, Roman law shaped continental civil procedure, and feudal law regulated obligations between lords and vassals.

This church legal tradition crystallized in the 12th century with Gratian’s Decretum and papal decretals, forming a sophisticated body of rules taught in universities. Its courts used Latin and often inquisitorial procedure, influencing European legal culture. Notably, “benefit of clergy” let clerics (and sometimes literate laymen) be tried there, and its machinery powered efforts against heresy, including the medieval Inquisition.