Yes. It belongs to both the Son and to the Holy Ghost to be invisibly sent because it belongs to both to dwell in the soul by grace.
The whole Trinity dwells in the mind by sanctifying grace, according to Jn. 14:23: "We will come to him, and will make Our abode with him." But that a divine person be sent to anyone by invisible grace signifies both that this person dwells in a new way within him and that He has His origin from another.
The soul is made like to God by grace. Hence for a divine person to be sent to anyone by grace, there must needs be a likening of the soul to the divine person Who is sent, by some gift of grace. Because the Holy Ghost is Love, the soul is assimilated to the Holy Ghost by the gift of charity: hence the mission of the Holy Ghost is according to the mode of charity. Whereas the Son is the Word, not any sort of word, but one Who breathes forth Love. Hence Augustine says (De Trin. ix 10): "The Word we speak of is knowledge with love."