The Spirit as the consummation of the processed Triune God is the Paraclete, the Comforter, to the believers (John 14:16-17). In John 14:16 the Lord said He would ask the Father to give us another Comforter. The Comforter is the Paraclete. The Greek word here means "advocate, one alongside who takes care of our cause, our affairs." The Spirit who is the life-giving Spirit, the Spirit of Jesus, the Spirit of Christ, the Spirit of Jesus Christ, and the Lord Spirit is the Paraclete to us, the One who is always alongside of us to take care of our case, our affairs, and our needs.
The life-giving Spirit is the reality of the processed Triune God (John 14:17a; 15:26b; 16:13; 1 John 5:6b). If we have this Spirit, we have the reality of the Divine Trinity, who is processed and consummated.
The life-giving Spirit is the reaching of the processed Triune God to the believers. If there were not such a Spirit, the Triune God would have no way to reach us. The Father is the source; the Son is the course; and the Spirit is the reaching. The Triune God reaches us through the Spirit that gives life.
The Spirit is also the believers' access unto the Father (Eph. 2:18), the source of the Divine Trinity. For the Spirit to be the "reaching" means that He comes to us. For Him to be the "access" means that we go to Him. This is the proper communication coming and going. This Spirit is God's coming, and this Spirit is also our entrance, the access. Without the Spirit, there is no entry, no access, for us to enter into the Triune God. For the Triune God to reach us, we need the Spirit. For us to enter into the Triune God, we need the Spirit. This Spirit is God's reaching; He is also our entry, our access, into the Divine Trinity.
Second Corinthians 13:14 reveals that the Spirit is the fellowship of the processed Triune God with the believers for their enjoyment of the riches of the Divine Trinity. When He reaches us, and we enter into Him, there is the fellowship. Then we remain in the fellowship to enjoy all the riches of the Triune God embodied in Christ.