From St. Thomas Aquinas’ Summa Theologiae, 3 q. 2 a. 6:
…we must know that two heresies have arisen with regard to the mystery of the union of the two natures in Christ. The first confused the natures, as Eutyches and Dioscorus, who held that from the two natures one nature resulted, so that they confessed Christ to be “from” two natures (which were distinct before the union), but not “in” two natures (the distinction of nature coming to an end after the union). The second was the heresy of Nestorius and Theodore of Mopsuestia, who separated the persons. For they held the Person of the Son of God to be distinct from the Person of the Son of man, and said these were mutually united: first, “by indwelling,” inasmuch as the Word of God dwelt in the man, as in a temple; secondly, “by unity of intention,” inasmuch as the will of the man was always in agreement with the will of the Word of God; thirdly, “by operation,” inasmuch as they said the man was the instrument of the Word of God; fourthly, “by greatness of honour,” inasmuch as all honour shown to the Son of God was equally shown to the Son of man, on account of His union with the Son of God; fifthly, “by equivocation,” i.e. communication of names, inasmuch as we say that this man is God and the Son of God. Now it is plain that these modes imply an accidental union.
But some more recent masters, thinking to avoid these heresies, through ignorance fell into them. For some conceded one person in Christ, but maintained two hypostases, or two supposita, saying that a man, composed of body and soul, was from the beginning of his conception assumed by the Word of God. And this is the first opinion set down by the Master (Sent. iii, D, 6). But others desirous of keeping the unity of person, held that the soul of Christ was not united to the body, but that these two were mutually separate, and were united to the Word accidentally, so that the number of persons might not be increased. And this is the third opinion which the Master sets down (Sent. iii, D, 6).
But both of these opinions fall into the heresy of Nestorius; the first, indeed, because to maintain two hypostases or supposita in Christ is the same as to maintain two persons, as was shown above (Article 3). And if stress is laid on the word “person,” we must have in mind that even Nestorius spoke of unity of person on account of the unity of dignity and honour. Hence the fifth Council (Constantinople II, coll. viii, can. 5) directs an anathema against such a one as holds “one person in dignity, honour and adoration, as Theodore and Nestorius foolishly wrote.” But the other opinion falls into the error of Nestorius by maintaining an accidental union. For there is no difference in saying that the Word of God is united to the Man Christ by indwelling, as in His temple (as Nestorius said), or by putting on man, as a garment, which is the third opinion; rather it says something worse than Nestorius–to wit, that the soul and body are not united.
Now the Catholic faith, holding the mean between the aforesaid positions, does not affirm that the union of God and man took place in the essence or nature, nor yet in something accidental, but midway, in a subsistence or hypostasis. Hence in the fifth Council (Constantinople II, coll. viii, can. 5) we read: “Since the unity may be understood in many ways, those who follow the impiety of Apollinaris and Eutyches, professing the destruction of what came together” (i.e. destroying both natures), “confess a union by mingling; but the followers of Theodore and Nestorius, maintaining division, introduce a union of purpose. But the Holy Church of God, rejecting the impiety of both these treasons, confesses a union of the Word of God with flesh, by composition, which is in subsistence.” Therefore it is plain that the second of the three opinions, mentioned by the Master (Sent. iii, D, 6), which holds one hypostasis of God and man, is not to be called an opinion, but an article of Catholic faith. So likewise the first opinion which holds two hypostases, and the third which holds an accidental union, are not to be styled opinions, but heresies condemned by the Church in Councils.
It can be shown that Islam is, in many ways, Nestorianism taken to its logical conclusion, and in fact the background of Islam has Nestorian influence. So these questions are far from having mere academic significance.