Cyril of Alexandria's dispute with Nestorius turned on a single word, but the word carried a whole Christology behind it. Nestorius, trained in the Antiochene tradition, would call Mary Christotokos, Christ-bearer, comfortably, but hesitated over Theotokos, God-bearer, worried it implied that the eternal, impassible God had been born and could suffer as a human infant. Cyril insisted on Theotokos precisely because he held that the divine Word and the human Jesus were not two separate subjects loosely joined but a single united reality, one hypostasis bearing both a divine and a human nature inseparably; if that union were real, then whatever could be said of the man Jesus, including being born of Mary, could be said of God the Word without qualification.
The Council of Ephesus sided with Cyril in 431, and Theotokos survives as a title of Mary across the Orthodox, Catholic and Oriental Orthodox traditions to this day, a rare point where Alexandria's Christology carried the field outright. But the underlying question, exactly how tightly the two natures were united within that single hypostasis, kept the controversy alive for another twenty years; when the Council of Chalcedon settled it in 451 with language of two natures united in one person, the Alexandrian tradition itself split over whether Chalcedon's formula preserved Cyril's unity closely enough, a division that produced the Oriental Orthodox churches and has never fully closed.