About 260 Bishop Dionysius of Alexandria wrote his letter to Ammonius and Euphranor against Sabellianism in which he expressed himself with inexactness as to the Logos and its relation to God the Father. Upon this an accusation against him was laid before Pope Dionysius who called a synod at Rome about 260 for the settlement of the matter. The pope issued, in his own name and that of the council, an important doctrinal letter in which, first, the erroneous doctrine of Sabellius was again condemned and, then, the false opinions of those were rejected who, like the Marcionites, in a similar manner separate the Divine monarchy into three entirely distinct hypostases or who represent the Son of God as a created being, while the Holy Scriptures declare Him to have been begotten passages in the Bible, such as Deut., xxxii, 6, Prov., viii, 22, cannot be cited in support of false doctrines such as these. Along with this doctrinal epistle Pope Dionysius sent a separate letter to the Alexandrian Bishop in which the latter was called on to explain his views. This Dionysius of Alexandria did in his "Apologia". According to the ancient practice of the Roman Church Dionysius also extended his care to the faithful of distant lands. When the Christians of Cappadocia were in great distress from the marauding incursions of the Goths, the pope addressed a consolatory letter to the Church of Caesarea and sent a large sum of money by messengers for the redemption of enslaved Christians. The great synod of Antioch which deposed Paul of Samosata sent a circular letter to Pope Dionysius and Bishop Maximus of Alexandria concerning its proceedings. After death the body of Dionysius was buried in the papal crypt in the catacomb of Callistus.
None of this implies the Supremacy of Jurisdiction. Primacy of Honor, definitely, tremendous respect for the Pope, which at even at that early time was well known as a bastion of orthodoxy, yes, but jurisdiction, no.