According to Catholicism, how can the Holy Spirit proceed from then Father and the Son if it is God?
Before beginning, let us make use of Sacred Scriptures to show the Holy Spirit proceeds from the Father and the Son.
How can we use Scripture to show the Holy Spirit proceeds from the Father and the Son?
One proof is that the Holy Spirit is referred to in Scripture as both the Spirit of the Father (Mt 10:20, Rom 8:10-11, 2 Cor 1:21-22, Eph 3:14-16) and as the Spirit of the Son (Rom 8:9, Gal 4:6, Phil 1:19, 1 Pt 1:11). Statements saying that the Spirit is “of” the other two Persons of the Trinity indicate that his Person is tightly bound up with and originates from them (just as the Son is the Son “of” the Father).
A second proof is that the external relations of the Trinity model their internal ones. In John 14:26 the Spirit is said to proceed from the Father, but a chapter later, in 15:26, Jesus states that he will send the Spirit from the Father. The same relation is reflected in Acts 2:33, where Peter states that Jesus has received the Spirit from the Father and sends him.
A philosophical explanation of this is found in the Council of Florence, which stated in 1439, “Since the Father has through generation given to the only-begotten Son everything that belongs to the Father, except being Father, the Son has also eternally from the Father, from whom he is eternally born, that the Holy Spirit proceeds from the Son” (Decree for the Greeks).
The Spirit proceeds from the Father and the Son because the Father has given all things to the Son, including the procession of the Holy Spirit. For more information see the Catechism of the Catholic Church 246-248, 264.
The Catechism of the Catholic Church simply declares that the Church believes that the Holy Spirit proceeds from then Father and the Son, without explaining how! It does give us glimmers to help us understand the Divine Mystery. Being a Divine Mystery, it will never be fully understood while in this life. Even in heaven, the Angels and Saints will be able to contemplate this Divine Mystery for all eternity and never get bored of it.
246 The Latin tradition of the Creed confesses that the Spirit "proceeds from the Father and the Son (filioque)". The Council of Florence in 1438 explains: "The Holy Spirit is eternally from Father and Son; He has his nature and subsistence at once (simul) from the Father and the Son. He proceeds eternally from both as from one principle and through one spiration. . . . And, since the Father has through generation given to the only-begotten Son everything that belongs to the Father, except being Father, the Son has also eternally from the Father, from whom he is eternally born, that the Holy Spirit proceeds from the Son.”
247 The affirmation of the filioque does not appear in the Creed confessed in 381 at Constantinople. But Pope St. Leo I, following an ancient Latin and Alexandrian tradition, had already confessed it dogmatically in 447,76 even before Rome, in 451 at the Council of Chalcedon, came to recognize and receive the Symbol of 381. The use of this formula in the Creed was gradually admitted into the Latin liturgy (between the eighth and eleventh centuries). The introduction of the filioque into the Niceno-Constantinopolitan Creed by the Latin liturgy constitutes moreover, even today, a point of disagreement with the Orthodox Churches.
248 At the outset the Eastern tradition expresses the Father's character as first origin of the Spirit. By confessing the Spirit as he "who proceeds from the Father", it affirms that he comes from the Father through the Son.77 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",78 for the eternal order of the divine persons in their consubstantial communion implies that the Father, as "the principle without principle",79 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.80 This legitimate complementarity, provided it does not become rigid, does not affect the identity of faith in the reality of the same mystery confessed.
264 "The Holy Spirit proceeds from the Father as the first principle and, by the eternal gift of this to the Son, from the communion of both the Father and the Son" (St. Augustine, De Trin. 15, 26, 47: PL 42, 1095).
The great Doctor of the Church, St. Thomas Aquinas in his Summa Theologiae, Question 27 and Question 36 addresses this issue: Spiration.
Whether the Holy Ghost proceeds from the Son?
It must be said that the Holy Ghost is from the Son. For if He were not from Him, He could in no wise be personally distinguished from Him; as appears from what has been said above (I:28:3; I:30:2). For it cannot be said that the divine Persons are distinguished from each other in any absolute sense; for it would follow that there would not be one essence of the three persons: since everything that is spoken of God in an absolute sense, belongs to the unity of essence. Therefore it must be said that the divine persons are distinguished from each other only by the relations. Now the relations cannot distinguish the persons except forasmuch as they are opposite relations; which appears from the fact that the Father has two relations, by one of which He is related to the Son, and by the other to the Holy Ghost; but these are not opposite relations, and therefore they do not make two persons, but belong only to the one person of the Father. If therefore in the Son and the Holy Ghost there were two relations only, whereby each of them were related to the Father, these relations would not be opposite to each other, as neither would be the two relations whereby the Father is related to them. Hence, as the person of the Father is one, it would follow that the person of the Son and of the Holy Ghost would be one, having two relations opposed to the two relations of the Father. But this is heretical since it destroys the Faith in the Trinity. Therefore the Son and the Holy Ghost must be related to each other by opposite relations. Now there cannot be in God any relations opposed to each other, except relations of origin, as proved above (I:28:44). And opposite relations of origin are to be understood as of a "principle," and of what is "from the principle." Therefore we must conclude that it is necessary to say that either the Son is from the Holy Ghost; which no one says; or that the Holy Ghost is from the Son, as we confess.
Furthermore, the order of the procession of each one agrees with this conclusion. For it was said above (I:27:4; I:28:4), that the Son proceeds by the way of the intellect as Word, and the Holy Ghost by way of the will as Love. Now love must proceed from a word. For we do not love anything unless we apprehend it by a mental conception. Hence also in this way it is manifest that the Holy Ghost proceeds from the Son.
We derive a knowledge of the same truth from the very order of nature itself. For we nowhere find that several things proceed from one without order except in those which differ only by their matter; as for instance one smith produces many knives distinct from each other materially, with no order to each other; whereas in things in which there is not only a material distinction we always find that some order exists in the multitude produced. Hence also in the order of creatures produced, the beauty of the divine wisdom is displayed. So if from the one Person of the Father, two persons proceed, the Son and the Holy Ghost, there must be some order between them. Nor can any other be assigned except the order of their nature, whereby one is from the other. Therefore it cannot be said that the Son and the Holy Ghost proceed from the Father in such a way as that neither of them proceeds from the other, unless we admit in them a material distinction; which is impossible.
Hence also the Greeks themselves recognize that the procession of the Holy Ghost has some order to the Son. For they grant that the Holy Ghost is the Spirit "of the Son"; and that He is from the Father "through the Son." Some of them are said also to concede that "He is from the Son"; or that "He flows from the Son," but not that He proceeds; which seems to come from ignorance or obstinacy. For a just consideration of the truth will convince anyone that the word procession is the one most commonly applied to all that denotes origin of any kind. For we use the term to describe any kind of origin; as when we say that a line proceeds from a point, a ray from the sun, a stream from a source, and likewise in everything else. Hence, granted that the Holy Ghost originates in any way from the Son, we can conclude that the Holy Ghost proceeds from the Son.
Whether the Holy Ghost proceeds from the Father through the Son?
Whenever one is said to act through another, this preposition "through" points out, in what is covered by it, some cause or principle of that act. But since action is a mean between the agent and the thing done, sometimes that which is covered by the preposition "through" is the cause of the action, as proceeding from the agent; and in that case it is the cause of why the agent acts, whether it be a final cause or a formal cause, whether it be effective or motive. It is a final cause when we say, for instance, that the artisan works through love of gain. It is a formal cause when we say that he works through his art. It is a motive cause when we say that he works through the command of another. Sometimes, however, that which is covered by this preposition "through" is the cause of the action regarded as terminated in the thing done; as, for instance, when we say, the artisan acts through the mallet, for this does not mean that the mallet is the cause why the artisan acts, but that it is the cause why the thing made proceeds from the artisan, and that it has even this effect from the artisan. This is why it is sometimes said that this preposition "through" sometimes denotes direct authority, as when we say, the king works through the bailiff; and sometimes indirect authority, as when we say, the bailiff works through the king.
Therefore, because the Son receives from the Father that the Holy Ghost proceeds from Him, it can be said that the Father spirates the Holy Ghost through the Son, or that the Holy Ghost proceeds from the Father through the Son, which has the same meaning.
Whether the Father and the Son are one principle of the Holy Ghost?
The Father and the Son are in everything one, wherever there is no distinction between them of opposite relation. Hence since there is no relative opposition between them as the principle of the Holy Ghost it follows that the Father and the Son are one principle of the Holy Ghost.
Some, however, assert that this proposition is incorrect: "The Father and the Son are one principle of the Holy Ghost," because, they declare, since the word "principle" in the singular number does not signify "person," but "property," it must be taken as an adjective; and forasmuch as an adjective cannot be modified by another adjective, it cannot properly be said that the Father and the Son are one principle of the Holy Ghost unless one be taken as an adverb, so that the meaning should be: They are one principle—that is, in one and the same way. But then it might be equally right to say that the Father is two principles of the Son and of the Holy Ghost—namely, in two ways. Therefore, we must say that, although this word "principle" signifies a property, it does so after the manner of a substantive, as do the words "father" and "son" even in things created. Hence it takes its number from the form it signifies, like other substantives. Therefore, as the Father and the Son are one God, by reason of the unity of the form that is signified by this word "God"; so they are one principle of the Holy Ghost by reason of the unity of the property that is signified in this word "principle."