"The Spirit reveals the glory of the Son of God, which perhaps explains why He is several times called "the Spirit of Christ" (Rom. 8:9; Gal. 4:6; 1 Peter 1:11). Evidently He sustains a special relationship with the Lord Jesus Christ, but H. C. G. Moule is undoubtedly correct when he writes, "the emphasis of the words here (is) on the work of the Holy Ghost as the Revealer of Christ to the soul."
Should we ask why it is the Spirit who performs this work? Several answers may be given: the Spirit is one with Him in essential union (2 Cor. 12:13; 1 John 5:7); He dwelt in Christ, in fullness and without measure (John 3:34); He is granted to us on account of Christ's redeeming work (Gal. 3:13,14); He was sent by Christ to perform His special ministry (John 15:26); He is responsible for applying to men the benefits of Christ (Eph. 1:17); He manifests Christ's presence in the soul (John 14:16-18); and He is the great "promise" of Christ's gospel (Gal. 3:14).
It is important to observe that the Spirit's work is not, in any way, to enhance the Lord's glory, but simply to throw light upon it, bringing it wonderfully into view. In this respect, it is akin to what happens in the natural world. The scenery in some given place may be very beautiful, and the sun, when it comes out, does not intrinsically change or improve it; yet its light does make a difference, transfiguring it by bringing out more fully its delightful loveliness. As the psalmist says, "With thee is the fountain of life: in thy light shall we see light" (Ps. 36:9).
The Spirit is moved to exercise His ministry. What exactly motivates and encourages Him in this? Since Scripture refers to "the love of the Spirit" (Rom. 15:30), no doubt His own eternal love for the Son is a major factor, and "to have an opportunity of letting His love go forth towards Him, in an exuberance of joy and affection is only what we might expect. He glorifies Christ." In particular, it is evident that the Spirit is lovingly concerned to redress the wrong done to the Son in the time of His humiliation, when, "despised and rejected of men" (Isa. 53:3), He was held up to dreadful indignity and shame (Matt. 26:67; Gal. 3:13; Heb. 12:2). The Spirit has even been the One to vindicate the Lord Jesus Christ (see 1 Tim. 3:16-"justified in the Spirit"). pg. 41-43