Though not a Catholic, something @MattGutting said struck me as particularly important and pregnant (no pun intended) with meaning. He said,
"Thus, God the Father is THE Father; the Son proceeds from him, and the Spirit from both. St. Thomas Aquinas, in the Summa Theologica, gives a philosophically dense discussion of the topic in the 'Treatise on the Most Holy Trinity' (First Part of the Summa, Questions 27–43). In Question 27, Article 1, he discusses what he calls 'procession' in God. He gives the analogy of a thought, or a word, of a person. The thought remains in the person, but at the same time can be thought of as separate from the person. It is in this sort of sense that the Son "proceeds from" the Father . . .."
After this quotation, @MattGutting inserts a quotation from Aquinas's Summa Theologica which contains a key word; namely, similitude.
Similitude, from a rhetorical point of view, is a kind of intellectual argument that appeals to similarity. Similes and metaphors are the most common and noticeable building blocks (or foundations) of arguments from similitude. For a more detailed look at this concept, see Richard Weaver's book The Ethics of Rhetoric (and for a brief introduction to it, go here).
When attempting to grasp the interrelationships within the Holy Trinity, we almost of necessity have to resort to similitude, as in the simile "the Holy Spirit is like ______," or the metaphor, "The Holy Spirit is the gasoline, as it were, that runs the engine of the Father, while the Son is the actual car in motion." Obviously, this kind of statement is patently absurd, but I use it for strictly illustrative purposes.
The concept of fuel/gas, however, is perhaps quite apt vis á vis the function of Holy Spirit within the Godhead, in general, and within Jesus, in particular. Fuel, of course, is the source of power, once it is ignited. In other words, fuel is a metaphor for motivation, for example.
Motivation, it has been suggested, is the act of "throwing oneself out of order and then achieving order once again" (look here for background information on Louis Tice, founder of the Pacific Institute, to whom I am indebted for this definition of motivation). This out-of-order phenomenon is characteristic of human behavior when a person's past behavior has to give way to a new behavior because of a change in goals.
Being a flesh-and-blood human being in space and time, Jesus was not exempt from this phenomenon. When he had completed one particular aspect of his God-given ministry, he was motivated to embark on another aspect of ministry. He succeeded in each and every task precisely because he was led, filled, impelled, anointed, and empowered by the Holy Spirit at each stage in his public--and private--life. Never was there a lapse in Spirit-controlled thought, word, or deed. Jesus could say of a truth that the food which sustained him was doing the will of his Father (see John 4:34).
In the life of Jesus, then, we notice that he was "motivated" to go from one place to another at the instigation of the Holy Spirit. After Jesus' baptism by John and just prior to Jesus' 40-day solitary sojourn in the wilderness wasteland, Matthew tells us Jesus was
". . . full of the Holy Spirit, [and he] returned from the Jordan [River] and was led up by [or led around by] the Spirit into the wilderness to be tempted by the devil" (4:1 NASB Updated, my italics).
Mark adds this thought:
"Immediately [after his baptism by John] the Spirit impelled Him to go out into the wilderness" (1:12 NASB Updated, my italics)
John adds this thought:
"'I have seen the Spirit descending as a dove out of heaven, and He remained upon Him'" (1:32, my italics).
Putting these verses together, we find that the Holy Spirit of God
Remained on Jesus
These are all "action words," and for there to be an action, there needs to be an expenditure of energy. Luke tells us in 4:14,
"And Jesus returned to Galilee in the power of the Spirit, and news about Him spread through all the surrounding district" (my italics).
We can therefore add a fifth action word--power--to the list of four, above. Jesus was empowered or energized by the Holy Spirit to do what his Father wanted him to do; namely, to preach and teach the gospel, which the prophet Isaiah prophesied and which Jesus fulfilled:
"THE SPIRIT OF THE LORD IS UPON ME, BECAUSE HE ANOINTED ME TO PREACH THE GOSPEL TO THE POOR. HE HAS SENT ME TO PROCLAIM RELEASE TO THE CAPTIVES, AND RECOVERY OF SIGHT TO THE BLIND, TO SET FREE THOSE WHO ARE OPPRESSED, TO PROCLAIM THE FAVORABLE YEAR OF THE LORD" (Luke 4:18-19 NASB Updated).
The above quotation, which Jesus read from the scroll of Isaiah in the synagogue in Nazareth on the Sabbath day, gives us a sixth word about the ministry of the Holy Spirit in Jesus' life and ministry; namely, anointed. How fitting that the Holy-Spirit-anointed Son of God should be called Messiah. As Constable observed,
The name Christ is the rough equivalent of the Hebrew “Messiah” or “Anointed One.” In the Old Testament it refers generally to people anointed for a special purpose including priests, kings, the patriarchs (metaphorically), and even the pagan king Cyrus. It came to have particular reference to the King whom God would provide from David’s line who would rule over Israel and the nations eventually (cf. 2 Sam. 7:12-16; Ps. 2:2: 105:15; et al.). The early Christians believed that Jesus of Nazareth was the Christ of the Old Testament. Because they used both names together, “Christ” became a virtual name for Jesus, a titulary (title turned name). Paul, for example, used it this way frequently in his writings.
In conclusion, as you can tell by now, the Holy Spirit, the Son, and the Father have what could fairly be called distinctive roles in the unfolding of salvation history from eternity to eternity. If we expect to comprehend--in this life, at any rate--how the three persons of the Godhead function as one and yet at the same time maintain a distinctness of roles, our expectations will be dashed every time. We need, therefore, to tread lightly and reverently, with feet bereft of shoes (see Exodus 3:5) as we contemplate what is essentially ineffable.