Do we shape the Holy Spirit?

Oxford, intermingle:

to mix people, ideas, colours, etc. together; to be mixed in this way.

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Prof NT Wright described the indwelling of the Holy Spirit in our human spirit, saying:

There is a kind of extraordinary intermingling of our spirit, our innermost breathing life, with God's own breath.

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Emphasis added.

In the first place, our spirit is an uncreated, detached breath of God that animates our soul. When we are born of the Spirit (born again), the Paraclete/Spirit takes up residence in our human spirit. This is not an intermingling of two different substances but a merging of the same kind of spiritual substance.

He said:

In shaping and directing the believer, … the Spirit's own self, having come home to you and me, and each one, has itself been shaped afresh.

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The Indwelling of the Spirit is to affect us, not the other way around. We do not shape the Spirit.

Just as the Incarnate Son was shaped as a human being by his friendships with Peter, James, John, Mary Magdalene. He became the human being that he was and is through those relationships. So the Spirit, having dwelt within us, … think the effect of that on the Spirit's own self.

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This analogical argument doesn't work. Wright anthropomorphized the Spirit too much. Jesus was a man. The Spirit isn't. Christians are called to partner with the Holy Spirit in living out their faith. "Work out your own salvation with fear and trembling, for it is God who works in you, both to will and to work for his good pleasure" (Ph 2:12). Human effort aligns with the Spirit’s transformative work. He shapes us.

At our death, God will download our software onto his hardware

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Actually, the term 'upload 'is more appropriate. When we die, our soul data/info will be uploaded to God's spiritual hardware for safekeeping.

until he gives us new hardware to run the software again for ourselves.

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Right, that's our resurrected bodies.

Do we shape the Holy Spirit?

No. The Holy Spirit is divine, sovereign, and immutable. I wouldn't personify the Spirit too much.