It may sound like a “process,” but since God does not exist in time the generation of the Persons of the Trinity takes no time but is an eternal event always occurring in the now. It takes no time for the Father to beget the Son. It takes no time for the Spirit to proceed from the Father and Son. It is all simultaneous and eternal. There is no before and after. Neither space nor time are applicable to God in his essence.
My first response is that it is Trinity being described in terms a human can partially understand, but that the relationship between the Father, Son and Holy Spirit cannot be fully understood by us mortals… ?
The Trinity has always been there - the Gospel of John Chapter 1. The word (Jesus) had not taken on flesh at the beginning but was always present as is the holy spirit. I don’t see a time issue here. As Jesus says I am in the father and the father is in me. As someone says I don’t think we can truly comprehend the Trinity until we reach heaven it is beyond us .
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