The Trinity in the Incarnation

The Father sent the Son to take up the great work of incarnation and propitiation, while he sent the Holy Spirit to make that work possible in the first place (as we saw above, through the virginal conception and the anointing of the man Jesus Christ), and then realize and complete the work after Christ had accomplished it...So the incarnate Son became a human person...he was the eternal divine person who took on human nature. Incarnational theology is a deep subject and is too complex to explore in detail here. But the classic term used in that doctrine is hypostatic union, which means that Jesus was one person who possessed both God's nature and man's nature. The Son became human. The Holy Spirit, on the other hand, did not take human nature into personal nature into personal union with himself. He did not become human. Instead, he indwells people.

Fred Sanders,The Deep Things of God