The Holy Spirit is Called the Gift of God in the Scriptures. By the Gift of the Holy Spirit is Meant the Gift Which is the Holy Spirit. The Holy Spirit is Specially Called Love, Although Not Only the Holy Spirit in the Trinity is Love
33 Is this too to be proved, that the Holy Spirit is called in the sacred books the gift of God? If people look for this too, we have in the Gospel according to John the words of our Lord Jesus Christ, who says, “If any one thirst, let him come to me and drink: he that believes in me, as the Scripture says, out of his belly shall flow rivers of living water.” And the evangelist has gone on further to add, “And this He spoke of the Spirit, which they should receive who believe in Him.” And hence Paul the apostle also says, “And we have all been made to drink into one Spirit.” The question then is, whether that water is called the gift of God which is the Holy Spirit. But as we find here that this water is the Holy Spirit, so we find elsewhere in the Gospel itself that this water is called the gift of God. For when the same Lord was talking with the woman of Samaria at the well, to whom He had said, “Give me to drink,” and she had answered that the Jews “have no dealings” with the Samaritans, Jesus answered and said unto her, “If you had known the gift of God, and who it is that says to you, Give me to drink, you would have asked of Him, and He would have given you living water. The woman says unto Him, Sir, you have nothing to draw with, and the well is deep: whence then have you this living water, etc.? Jesus answered and said unto her, Every one that drinks of this water shall thirst again; but whose shall drink of the water that I shall give him, shall never thirst; but the water that I shall give him, shall be in him a fountain of water springing up unto eternal life.” Because this living water, then, as the evangelist has explained to us, is the Holy Spirit, without doubt the Spirit is the gift of God, of which the Lord says here, “If you had known the gift of God, and who it is that says unto you, Give me to drink, you would have asked of Him, and He would have given you living water.” For that which is in the one passage, “Out of his belly shall flow rivers of living water,” is in the other, “shall be in him a fountain of water springing up unto eternal life.”
34. Paul the apostle also says, “To each of us is given grace according to the measure of the gift of Christ;” and then, that he might show that by the gift of Christ he meant the Holy Spirit, he has gone on to add, “Wherefore He says, He has ascended up on high, He has led captivity captive, and has given gifts to men.” And every one knows that the Lord Jesus, when He had ascended into heaven after the resurrection from the dead, gave the Holy Spirit, with whom they who believed were filled, and spoke with the tongues of all nations. And let no one object that he says gifts, not gift: for he quoted the text from the Psalm. And in the Psalm it is read thus, “You have ascended up on high, You have led captivity captive, You have received gifts in men.” For so it stands in many mss., especially in the Greek mss., and so we have it translated from the Hebrew. The apostle therefore said gifts, as the prophet did, not gift. But whereas the prophet said, “You have received gifts in men,” the apostle has preferred saying, “He gave gifts to men:” and this in order that the fullest sense may be gathered from both expressions, the one prophetic, the other apostolic; because both possess the authority of a divine utterance. For both are true, as well that He gave to men, as that He received in men. He gave to men, as the head to His own members: He Himself that gave, received in men, no doubt as in His own members; on account of which, namely, His own members, He cried from heaven, “Saul, Saul, why do you persecute me?” And of which, namely, His own members, He says, “Since you have done it to one of the least of these that are mine, you have done it unto me.” Christ Himself, therefore, both gave from heaven and received on earth. And further, both prophet and apostle have said gifts for this reason, because many gifts, which are proper to each, are divided in common to all the members of Christ, by the Gift, which is the Holy Spirit. For each severally has not all, but some have these and some have those; although all have the Gift itself by which that which is proper to each is divided to Him, i.e. the Holy Spirit. For elsewhere also, when he had mentioned many gifts, “All these,” he says, “works that one and the self-same Spirit, dividing to each severally as He will.” And this word is found also in the Epistle to the Hebrews, where it is written, “God also bearing witness both with signs and wonders, and with various miracles, and gifts of the Holy Ghost.” And so here, when he had said, “He ascended up on high, He led captivity captive, He gave gifts to men,” he says further, “But that He ascended, what is it but that He also first descended into the lower parts of the earth? He who descended is the same also that ascended up far above all heavens, that He might fill all things. And He gave some apostles, some prophets, and some evangelists, and some pastors and doctors.” (This we see is the reason why gifts are spoken of; because, as he says elsewhere, “Are all apostles? Are all prophets?” etc.) And here he has added, “For the perfecting of the saints, for the work of the ministry, for the building up of the body of Christ.” This is the house which, as the Psalm sings, is built up after the captivity; since the house of Christ, which house is called His Church, is built up of those who have been rescued from the devil, by whom they were held captive. But He Himself led this captivity captive, who conquered the devil. And that he might not draw with him into eternal punishment those who were to become the members of the Holy Head, He bound him first by the bonds of righteousness, and then by those of might. The devil himself, therefore, is called captivity, which He led captive who ascended up on high, and gave gifts to men, or received gifts in men.
Source: On the Holy Trinity (New Advent)