The Father alone is the source of the Holy Spirit. The Father is the first origin of the Holy Spirit. The Holy Spirit is the Spirit of Christ from eternity.
And while in the property of each Person the Father is one, the Son is another, and the Holy Ghost is another, yet the Godhead is not distinct and different; for while the Son is the Only begotten of the Father, the Holy Spirit is the Spirit of the Father and the Son, not in the way that every creature is the creature of the Father and the Son, but as living and having power with Both, and
eternally subsisting of That Which is the Father and the Son.
The Holy Spirit does not have two sources but one source, the Father.
The Latins asserted that they say the holy Spirit proceeds from the Father and the Son not with the intention of excluding the
Father from being the source and principle of all deity, that is of the Son and of the holy Spirit, nor to imply that the Son does not receive from the Father, because the holy Spirit proceeds from the Son, nor that they posit two principles or two spirations; but they assert that there is only one principle and a single spiration of the holy Spirit, as they have asserted hitherto.
The Catechism of the Catholic Church:
248 The Western tradition expresses first the consubstantial communion between Father and Son, by saying that the Spirit proceeds from the Father and the Son (filioque). It says this, “legitimately and with good reason,” for the eternal order of the divine persons in their consubstantial communion implies that
the Father, as “the principle without principle,”
is the first origin of the Spirit, but also that as Father of the only Son, he is, with the Son, the single principle from which the Holy Spirit proceeds.
St Cyril: Anathema IX - The Holy Spirit is Christ’s from eternity:
If any man shall say that the one Lord Jesus Christ was glorified by the Holy Ghost, so that he used through him a power not his own and from him received power against unclean spirits and power to work miracles before men and
shall not rather confess that it was his own Spirit through which he worked these divine signs;
let him be anathema.
Nestorius: Anathema IX - The Holy Spirit was not Christ’s until Christ was conceived:
If anyone says that the form of a servant is of like nature with the Holy Ghost, and not rather that it owes its union with the Word which has existed since the conception, to his mediation, by which it works miraculous healings among men, and possesses the power of expelling demons; let him be anathema.