To better understand the doctrine of supposita and the role it plays in the doctrines of Trinity and Incarnation, we turn to Alfred J. Freddoso, Human Nature, Potency and the Incarnation (bolding added):
According to the Christian faith, as defined in this instance by the great Christological Councils and mirrored in centuries of liturgical practice and theological reflection, Jesus Christ is truly God and truly man. More precisely, he is a single divine person, the eternal Word, in whom are united, whole and unmixed, a divine nature and a human nature–so that he is, to quote Chalcedon, "one with the Father in his divinity and one with us in his humanity."
In expounding this doctrine medieval theologians fashioned the technical metaphysical notion of a suppositum (or hypostasis), i.e. an independently existing ultimate subject of characteristics.4 The philosophically astute will detect at once that, so understood, the concept of a suppositum is remarkably akin to that of an Aristotelian primary substance or individual(ized) nature. Indeed, had it not been for the doctrines of the Trinity and the Incarnation, medieval Christian thinkers would never have been led to assert that suppositum and substance (or: individual nature) are distinct concepts. To speak now only of the Incarnation, Christ's individual human nature, i.e. the individual composed of a body and an intellective soul and united to the Son of God, is a paradigmatic Aristotelian [primary] substance . . . .Yet, because of its metaphysical union with and dependence upon the eternal Word, this nature is not the ultimate metaphysical subject of Christ's characteristics–not even of his "purely human" characteristics. So in this one instance, known to us only by divine revelation, we have a substance which is not a suppositum, a substance which is metaphysically "sustained" by something distinct from it. To complete the picture, a person is just a suppositum with an intellectual nature, i.e. a suppositum essentially endowed with intellect and free will.
In technical medieval terminology, then, Jesus Christ is a divine suppositum or person, the Son of God, who has freely "assumed" and now "sustains" an individual human nature. What's more, this human nature is united to the divine person "hypostatically," i.e. in such a way that properties had immediately by the human nature have the Son of God as their ultimate metaphysical subject–in a manner analogous to that in which many properties had immediately by a proper part of a whole have the whole itself as their ultimate metaphysical subject.
