Nestorius makes the assertion that there was two natures of the second person of the Trinity. He believes there is the divine nature and the human nature. Jesus is occupied by the divine Logos. The two natures achieve union by will because they agree upon Jesus’s time on earth.
Cyril argued that Jesus’ full divinity and full humanity was unified in one nature. He had political reasons to disagree with Nestorius because he felt he was losing power when Nestorius was made equal to the Bishop of Rome. This connects to the issue at the Council of Nicea regarding the issue of the Metropolian Bishops.
Cyril believe Mary was the Mother of God “theotokos” because there was only one nature, therefore she had to give birth to God. Nestorius believed in “Christokos” meaning Mary was the Mother of Christ, not God due to the two natures of Jesus. Mary could not give birth to God, only the human nature of Jesus.