Objection 1. It would seem that the union of Incarnate Word did not take place in the person. For the Person of God is not distinct from His Nature, as we said (I, 39, 1). If, therefore, the union did not take place in the nature, it follows that it did not take place in the person.
Objection 2. Further, Christ's human nature has no less dignity than ours. But personality belongs to dignity, as was stated above (I, 29, 3, ad 2). Hence, since our human nature has its proper personality, much more reason was there that Christ's should have its proper personality.
Objection 3. Further, as Boethius says (De Duab. Nat.), a person is an individual substance of rational nature. But the Word of God assumed an individual human nature, for "universal human nature does not exist of itself, but is the object of pure thought," as Damascene says (De Fide Orth. iii, 11). Therefore the human nature of Christ has its personality. Hence it does not seem that the union took place in the person.
On the contrary, We read in the Synod of Chalcedon (Part ii, act. 5): "We confess that our Lord Jesus Christ is not parted or divided into two persons, but is one and the same only-Begotten Son and Word of God." Therefore the union took place in the person.
I answer that, Person has a different meaning from "nature." For nature, as has been said (1), designates the specific essence which is signified by the definition. And if nothing was found to be added to what belongs to the notion of the species, there would be no need to distinguish the nature from the suppositum of the nature (which is the individual subsisting in this nature), because every individual subsisting in a nature would be altogether one with its nature. Now in certain subsisting things we happen to find what does not belong to the notion of the species, viz. accidents and individuating principles, which appears chiefly in such as are composed of matter and form. Hence in such as these the nature and the suppositum really differ; not indeed as if they were wholly separate, but because the suppositum includes the nature, and in addition certain other things outside the notion of the species. Hence the suppositum is taken to be a whole which has the nature as its formal part to perfect it; and consequently in such as are composed of matter and form the nature is not predicated of the suppositum, for we do not say that this man is his manhood. But if there is a thing in which there is nothing outside the species or its nature (as in God), the suppositum and the nature are not really distinct in it, but only in our way of thinking, inasmuch it is called "nature" as it is an essence, and a "suppositum" as it is subsisting. And what is said of a suppositum is to be applied to a person in rational or intellectual creatures; for a person is nothing else than "an individual substance of rational nature," according to Boethius. Therefore, whatever adheres to a person is united to it in person, whether it belongs to its nature or not. Hence, if the human nature is not united to God the Word in person, it is nowise united to Him; and thus belief in Incarnation is altogether done away with, and Christian faith wholly overturned. Therefore, inasmuch as the Word has a human nature united to Him, which does not belong to His Divine Nature, it follows that the union took place in the Person of the Word, and not in the nature.
Reply to Objection 1. Although in God Nature and Person are not really distinct, yet they have distinct meanings, as was said above, inasmuch as person signifies after the manner of something subsisting. And because human nature is united to the Word, so that the Word subsists in it, and not so that His Nature receives therefrom any addition or change, it follows that the union of human nature to the Word of God took place in the person, and not in the nature.
Reply to Objection 2. Personality pertains of necessity to the dignity of a thing, and to its perfection so far as it pertains to the dignity and perfection of that thing to exist by itself (which is understood by the word "person"). Now it is a greater dignity to exist in something nobler than oneself than to exist by oneself. Hence the human nature of Christ has a greater dignity than ours, from this very fact that in us, being existent by itself, it has its own personality, but in Christ it exists in the Person of the Word. Thus to perfect the species belongs to the dignity of a form, yet the sensitive part in man, on account of its union with the nobler form which perfects the species, is more noble than in brutes, where it is itself the form which perfects.
Reply to Objection 3. The Word of God "did not assume human nature in general, but 'in atomo'"--that is, in an individual--as Damascene says (De Fide Orth. iii, 11) otherwise every man would be the Word of God, even as Christ was. Yet we must bear in mind that not every individual in the genus of substance, even in rational nature, is a person, but that alone which exists by itself, and not that which exists in some more perfect thing. Hence the hand of Socrates, although it is a kind of individual, is not a person, because it does not exist by itself, but in something more perfect, viz. in the whole. And hence, too, this is signified by a "person" being defined as "an individual substance," for the hand is not a complete substance, but part of a substance. Therefore, although this human nature is a kind of individual in the genus of substance, it has not its own personality, because it does not exist separately, but in something more perfect, viz. in the Person of the Word. Therefore the union took place in the person.
Source: Summa Theologica (New Advent)