Because the Holy Spirit is the Spirit of the Father and the Spirit of the Son.
In “On the Trinity” (De Trinitate), Book XV, Ch. XXVI, St. Augustine wrote,
Furthermore, in that highest Trinity which is God, there are no intervals of time, by which it can be shown, or at least inquired, whether the Son was born of the Father first and then afterwards the Holy Spirit proceeded from both, since holy scripture calls him (i.e., the Holy Spirit) the Spirit of both.
For, it is he (i.e.., the Holy Spirit) of whom the apostle says (Gal. 4:6), “But because you are sons, God has sent forth the Spirit of His Son into your hearts,” and it is he (i.e., the Holy Spirit) of whom the same Son says (Matt. 10:20), “For it is not you who speak, but the Spirit of your Father who speaks in you.”
And it is proved by many other testimonies of the divine word, that the Spirit, who is specially called in the Trinity, the Holy Spirit, is the Father’s and the Son’s.
Deinde in illa summa Trinitate quae Deus est, intervalla temporum nulla sunt, per quae possit ostendi aut saltem requiri, utrum prius de Patre natus sit Filius, et postea de ambobus processerit Spiritus Sanctus. Quoniam Scriptura sancta Spiritum eum dicit amborum.
Ipse est enim de quo dicit Apostolus: Quoniam autem estis filii, misit Deus Spiritum Filii sui in corda vestra: et ipse est de quo dicit idem Filius: Non enim vos estis qui loquimini; sed Spiritus Patris vestri, qui loquitur in vobis.
Et multis aliis divinorum eloquiorum testimoniis comprobatur Patris et Filii esse Spiritum, qui proprie dicitur in Trinitate Spiritus Sanctus.
In “Catechesis of the Illuminated,” Lecture XVII, §4, St. Cyril of Jerusalem wrote,
For he is called “Spirit” according what is just read (1 Cor. 12:8), “For to one is given the word of wisdom by the Spirit.” And he is called “Spirit of truth,” just as the savior says (John 16:13), “But when that one comes, the Spirit of truth.” And he is called “Comforter,” just as it is said (John 16:7), “For if I do not go away, the Comforter will not come to you.” But that he is one and the same, while called by different appellations, is demonstrated from these. For, concerning the Holy Spirit and the Comforter being the same, it is said (John 14:26), “But the Comforter, the Holy Spirit…” And concerning the Comforter and the Spirit of truth being the same, it is said (John 14:16-17), “…and I shall give you another Comforter, so that he may abide with you forever, the Spirit of truth…” And again (John 15:26), “But when the Comforter comes, whom I shall send to you from the Father, the Spirit of truth…” He is also called “Spirit of God,” just as it is written (John 1:32), “And I saw the Spirit of God descending…” And again (Rom. 8:14), “For as many as are led by the Spirit of God, these are the sons of God.” He is also called “Spirit of the Father,” just as the Savior says (Matt. 10:20), “For it is not you who speak, but the Spirit of your Father who speaks in you.” And again, Paulos [says] (Eph. 3:14-16), “For this reason, I bend my knees before the Father, etc., …that He would grant you…to be strengthened with power by His Spirit…” He is also called “Spirit of the Lord,” just as Petros said (Acts 5:9), “Why is it that you have consorted to tempt the Spirit of the Lord?” He is also called “Spirit of God” and “[Spirit of] Christ”, just as Paulos writes (Rom. 8:9), “But you are not in the flesh, but rather, in the Spirit, if indeed the Spirit of God dwells in you. But if someone does not have the Spirit of Christ, this one is not his.” He is also called “Spirit of the Son of God,” just as it is said (Gal. 4:6), “But since you are sons, God has sent forth the Spirit of His Son…” He is also called “Spirit of Christ,” just as it is written (1 Pet. 1:11), “…for what [person] or what manner of time the Spirit of Christ in them was indicating…” And again (Phil. 1:19), “…by your prayer and the supply of the Spirit of Jesus Christ.”
Καλεῖται μὲν γὰρ πνεῦμα κατὰ τὸ ἀρτίως ἀνεγνωσμένον, ᾧ μὲν γὰρ διὰ τοῦ πνεύματος δίδοται λόγος σοφίας. καλεῖται δὲ πνεῦμα ἀληθείας, καθὼς ὁ σωτήρ φησιν, ὅταν δὲ ἔλθῃ ἐκεῖνος, τὸ πνεῦμα τῆς ἀληθείας. καλεῖται καὶ παράκλητος, καθὼς εἴρηκεν, ἐὰν γὰρ ἐγὼ μὴ ἀπέλθω, ὁ παράκλητος οὐ μὴ ἔλθῃ πρὸς ὑμᾶς. ὅτι δὲ ἓν καὶ τὸ αὐτό ἐστι διαφόροις ταῖς προσηγορίαις ὀνομαζόμενον, δείκνυται σαφῶς ἐκ τούτων. περὶ μὲν γὰρ τοῦ εἶναι τὸ αὐτὸ τὸ πνεῦμα τὸ ἅγιον καὶ τὸν παράκλητον εἴρηται ὁ δὲ παράκλητος τὸ πνεῦμα τὸ ἅγιον. περὶ δὲ τοῦ τὸ αὐτὸ εἶναι καὶ τὸν παράκλητον καὶ τὸ πνεῦμα τῆς ἀληθείας εἴρηται καὶ ἄλλον παράκλητον δώσω ὑμῖν, ἵνα μεθ’ ὑμῶν μένῃ εἰς τὸν αἰῶνα, τὸ πνεῦμα τῆς ἀληθείας. καὶ πάλιν ὅταν δὲ ἔλθῃ ὁ παράκλητος ὃν ἐγὼ πέμψω ὑμῖν παρὰ τοῦ πατρός, τὸ πνεῦμα τῆς ἀληθείας. Καλεῖται καὶ πνεῦμα θεοῦ, καθὼς γέγραπται· καὶ εἶδον τὸ πνεῦμα τοῦ θεοῦ καταβαῖνον, καὶ πάλιν, ὅσοι γὰρ πνεύματι θεοῦ ἄγονται, οὗτοι υἱοὶ θεοῦ εἰσιν. καλεῖται καὶ πνεῦμα πατρός, καθώς φησιν ὁ σωτήρ· οὐ γὰρ ὑμεῖς ἐστε οἱ λαλοῦντες, ἀλλὰ τὸ πνεῦμα τοῦ πατρὸς ὑμῶν τὸ λαλοῦν ἐν ὑμῖν. καὶ πάλιν ὁ Παῦλος· τούτου χάριν κάμπτω τὰ γόνατά μου πρὸς τὸν πατέρα, καὶ ἑξῆς, ἵνα δῷ ὑμῖν κραταιωθῆναι διὰ τοῦ πνεύματος αὐτοῦ. καλεῖται καὶ πνεῦμα κυρίου, καθὼς Πέτρος · τί ὅτι συνεφωνήθη ὑμῖν πειράσαι τὸ πνεῦμα κυρίου; καλεῖται καὶ πνεῦμα θεοῦ καὶ Χριστοῦ, καθὼς γράφει Παῦλος· ὑμεῖς δὲ οὐκ ἐστὲ ἐν σαρκί, ἀλλ’ ἐν πνεύματι, εἴπερ πνεῦμα θεοῦ οἰκεῖ ἐν ὑμῖν. εἰ δέ τις πνεῦμα Χριστοῦ οὐκ ἔχει, οὗτος οὐκ ἔστιν αὐτοῦ. καλεῖται καὶ πνεῦμα τοῦ υἱοῦ τοῦ θεοῦ, καθὼς εἴρηται· ὅτι δέ ἐστε υἱοί, ἐξαπέστειλεν ὁ θεὸς τὸ πνεῦμα τοῦ υἱοῦ αὐτοῦ. καλεῖται καὶ πνεῦμα Χριστοῦ, καθὼς γέγραπται· εἰς τίνα ἢ ποῖον καιρὸν ἐδήλου τὸ ἐν αὐτοῖς πνεῦμα Χριστοῦ. καὶ πάλιν· διὰ τῆς ὑμῶν δεήσεως καὶ ἐπιχορηγίας τοῦ πνεύματος Ἰησοῦ Χριστοῦ.
Since the Holy Spirit, the Spirit of the Father and the Son, dwells in Christians, the Christian is spiritually united with the Father and the Son, for "he who is joined to the Lord is one spirit" (1 Cor. 6:17).