Objection 1. It would seem that the Holy Ghost does not proceed from the Father through the Son. For whatever proceeds from one through another, does not proceed immediately. Therefore, if the Holy Ghost proceeds from the Father through the Son, He does not proceed immediately; which seems to be unfitting.
Objection 2. Further, if the Holy Ghost proceeds from the Father through the Son, He does not proceed from the Son, except on account of the Father. But "whatever causes a thing to be such is yet more so." Therefore He proceeds more from the Father than from the Son.
Objection 3. Further, the Son has His being by generation. Therefore if the Holy Ghost is from the Father through the Son, it follows that the Son is first generated and afterwards the Holy Ghost proceeds; and thus the procession of the Holy Ghost is not eternal, which is heretical.
Objection 4. Further, when anyone acts through another, the same may be said conversely. For as we say that the king acts through the bailiff, so it can be said conversely that the bailiff acts through the king. But we can never say that the Son spirates the Holy Ghost through the Father. Therefore it can never be said that the Father spirates the Holy Ghost through the Son.
On the contrary, Hilary says (De Trin. xii): "Keep me, I pray, in this expression of my faith, that I may ever possess the Father--namely Thyself: that I may adore Thy Son together with Thee: and that I may deserve Thy Holy Spirit, who is through Thy Only Begotten."
I answer that, 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.
Reply to Objection 1. In every action two things are to be considered, the "suppositum" acting, and the power whereby it acts; as, for instance, fire heats through heat. So if we consider in the Father and the Son the power whereby they spirate the Holy Ghost, there is no mean, for this is one and the same power. But if we consider the persons themselves spirating, then, as the Holy Ghost proceeds both from the Father and from the Son, the Holy Ghost proceeds from the Father immediately, as from Him, and mediately, as from the Son; and thus He is said to proceed from the Father through the Son. So also did Abel proceed immediately from Adam, inasmuch as Adam was his father; and mediately, as Eve was his mother, who proceeded from Adam; although, indeed, this example of a material procession is inept to signify the immaterial procession of the divine persons.
Reply to Objection 2. If the Son received from the Father a numerically distinct power for the spiration of the Holy Ghost, it would follow that He would be a secondary and instrumental cause; and thus the Holy Ghost would proceed more from the Father than from the Son; whereas, on the contrary, the same spirative power belongs to the Father and to the Son; and therefore the Holy Ghost proceeds equally from both, although sometimes He is said to proceed principally or properly from the Father, because the Son has this power from the Father.
Reply to Objection 3. As the begetting of the Son is co-eternal with the begetter (and hence the Father does not exist before begetting the Son), so the procession of the Holy Ghost is co-eternal with His principle. Hence, the Son was not begotten before the Holy Ghost proceeded; but each of the operations is eternal.
Reply to Objection 4. When anyone is said to work through anything, the converse proposition is not always true. For we do not say that the mallet works through the carpenter; whereas we can say that the bailiff acts through the king, because it is the bailiff's place to act, since he is master of his own act, but it is not the mallet's place to act, but only to be made to act, and hence it is used only as an instrument. The bailiff is, however, said to act through the king, although this preposition "through" denotes a medium, for the more a "suppositum" is prior in action, so much the more is its power immediate as regards the effect, inasmuch as the power of the first cause joins the second cause to its effect. Hence also first principles are said to be immediate in the demonstrative sciences. Therefore, so far as the bailiff is a medium according to the order of the subject's acting, the king is said to work through the bailiff; but according to the order of powers, the bailiff is said to act through the king, forasmuch as the power of the king gives the bailiff's action its effect. Now there is no order of power between Father and Son, but only order of 'supposita'; and hence we say that the Father spirates through the Son; and not conversely.
Source: Summa Theologica (New Advent)