6 Some one may here inquire whether the Holy Spirit proceeds also from the Son. For the Son is Son of the Father alone, and the Father is Father of the Son alone; but the Holy Spirit is not the Spirit of one of them, but of both. You have the Lord Himself saying, “For it is not ye that speak, but the Spirit of your Father that speaks in you;” and you have the apostle, “God has sent forth the spirit of His Son into your hearts.” Are there, then, two, the one of the Father, the other of the Son?
Certainly not. For there is “one body,” he said, when referring to the Church; and presently added, “and one Spirit.” And mark how he there makes up the Trinity. “As you are called,” he says, “in one hope of your calling.” “One Lord,” where he certainly meant Christ to be understood; but it remained that he should also name the Father: and accordingly there follows, “One faith, one baptism, one God and Father of all, who is above all, and through all, and in you all.” And since, then, just as there is one Father, and one Lord, namely, the Son, so also there is one Spirit; He is doubtless of both: especially as Christ Jesus Himself says, “The Spirit of your Father that dwells in you;” and the apostle declares, “God has sent forth the Spirit of His Son into your hearts.”
You have the same apostle saying in another place, “But if the Spirit of Him that raised up Jesus from the dead dwell in you,” where he certainly intended the Spirit of the Father to be understood; of whom, however, he says in another place, “But if any man have not the Spirit of Christ, he is none of His.” And many other testimonies there are, which plainly show that He, who in the Trinity is styled the Holy Spirit, is the Spirit both of the Father and of the Son.
Source: Tractates on the Gospel of John (New Advent)