In the tradition of the Church, we have several texts. Pope Leo XIII's Divino Munus Illud is a classic text. Indwelling is produced by the presence of the whole Trinity, nevertheless is attributed in a special way to the Holy Spirit. The fullness of divine gifts is a consequence of the dwelling of the Holy Spirit. Without these, there is no beginning to a good life, no arriving at eternal salvation.
Grace is thus critical.
Each divine person engages in a distinct relationship with the soul. Each divine person has its specific role and influence. The Trinity is always acting in unity. The divine indwelling is not metaphorical, but is equally real as in the Eucharist. In the latter, the species of the bread and wine do not love, though we can love.