Objection 1. It would seem that there cannot be any procession in God. For procession signifies outward movement. But in God there is nothing mobile, nor anything extraneous. Therefore neither is there procession in God.
Objection 2. Further, everything which proceeds differs from that whence it proceeds. But in God there is no diversity; but supreme simplicity. Therefore in God there is no procession.
Objection 3. Further, to proceed from another seems to be against the nature of the first principle. But God is the first principle, as shown above (Question 2, Article 3). Therefore in God there is no procession.
On the contrary, Our Lord says, "From God I proceeded" (John 8:42).
I answer that, Divine Scripture uses, in relation to God, names which signify procession. This procession has been differently understood. Some have understood it in the sense of an effect, proceeding from its cause; so Arius took it, saying that the Son proceeds from the Father as His primary creature, and that the Holy Ghost proceeds from the Father and the Son as the creature of both. In this sense neither the Son nor the Holy Ghost would be true God: and this is contrary to what is said of the Son, "That... we may be in His true Son. This is true God" (1 John 5:20). Of the Holy Ghost it is also said, "Know you not that your members are the temple of the Holy Ghost?" (1 Corinthians 6:19). Now, to have a temple is God's prerogative. Others take this procession to mean the cause proceeding to the effect, as moving it, or impressing its own likeness on it; in which sense it was understood by Sabellius, who said that God the Father is called Son in assuming flesh from the Virgin, and that the Father also is called Holy Ghost in sanctifying the rational creature, and moving it to life. The words of the Lord contradict such a meaning, when He speaks of Himself, "The Son cannot of Himself do anything" (John 5:19); while many other passages show the same, whereby we know that the Father is not the Son. Careful examination shows that both of these opinions take procession as meaning an outward act; hence neither of them affirms procession as existing in God Himself; whereas, since procession always supposes action, and as there is an outward procession corresponding to the act tending to external matter, so there must be an inward procession corresponding to the act remaining within the agent. This applies most conspicuously to the intellect, the action of which remains in the intelligent agent. For whenever we understand, by the very fact of understanding there proceeds something within us, which is a conception of the object understood, a conception issuing from our intellectual power and proceeding from our knowledge of that object. This conception is signified by the spoken word; and it is called the word of the heart signified by the word of the voice.
As God is above all things, we should understand what is said of God, not according to the mode of the lowest creatures, namely bodies, but from the similitude of the highest creatures, the intellectual substances; while even the similitudes derived from these fall short in the representation of divine objects. Procession, therefore, is not to be understood from what it is in bodies, either according to local movement or by way of a cause proceeding forth to its exterior effect, as, for instance, like heat from the agent to the thing made hot. Rather it is to be understood by way of an intelligible emanation, for example, of the intelligible word which proceeds from the speaker, yet remains in him. In that sense the Catholic Faith understands procession as existing in God.
Reply to Objection 1. This objection comes from the idea of procession in the sense of local motion, or of an action tending to external matter, or to an exterior effect; which kind of procession does not exist in God, as we have explained.
Reply to Objection 2. Whatever proceeds by way of outward procession is necessarily distinct from the source whence it proceeds, whereas, whatever proceeds within by an intelligible procession is not necessarily distinct; indeed, the more perfectly it proceeds, the more closely it is one with the source whence it proceeds. For it is clear that the more a thing is understood, the more closely is the intellectual conception joined and united to the intelligent agent; since the intellect by the very act of understanding is made one with the object understood. Thus, as the divine intelligence is the very supreme perfection of God (14, 2), the divine Word is of necessity perfectly one with the source whence He proceeds, without any kind of diversity.
Reply to Objection 3. To proceed from a principle, so as to be something outside and distinct from that principle, is irreconcilable with the idea of a first principle; whereas an intimate and uniform procession by way of an intelligible act is included in the idea of a first principle. For when we call the builder the principle of the house, in the idea of such a principle is included that of his art; and it would be included in the idea of the first principle were the builder the first principle of the house. God, Who is the first principle of all things, may be compared to things created as the architect is to things designed.
Source: Summa Theologica (New Advent)