"Spirit"of God the Father, Son and the Holy Spirit? Biblically there's is a phrase for the Spirit of God the Father and the Spirit of Christ the Son. And the Holy Spirit proceeds from the Father and the Son.

How about the "Spirit" of the Holy Spirit? Looking at it, it seems that, the thing called "spirit" do not point to the Three Person in the Triune God. The phrase would mean as if the Triune God has a "spirit" that animates the three person namely, Father Son and the Holy Spirit.

God is a Spirit, how come God who is a Spirit also has a "spirit"?

Comparing to a created man who is composed of body and soul. The soul animates the body and the spirit animates the soul.

Is God who existed in eternity as Spirit animated by "spirit" too?

What is the logical and biblical understanding of the "spirit" of the Father, Son and the Holy Spirit? 

Catholicism is preferred but a Christian biblical understanding is also welcome.
Jesus said :

Pneuma 'o Theos [John 4:24 TR]
Spirit the God  (literally)

This is an equivalence. He is saying that 'God' means 'Spirit'. The nature of deity is a spiritual nature. The scripture (written by Paul the apostle) also says (of the Son of God, Jesus Christ) :

... being in form, of God ... Philippians 2:6 (literal)

The form of God is spirit. God is a spirit.
This is a matter of the nature of Deity.
This nature is shared. From the beginning the plural (or 'composite') term Elohim is used in Genesis to express Deity. It is a matter of union. A perfection of unity within Deity.
The term 'Spirit of Christ' is used in scripture (I Peter 1:11, for example) and the term 'Spirit of the Father' is also used (Matthew 10:20, for example). But the term 'Spirit of the Holy Spirit' is never used.
Thus we learn that the Holy Spirit acts on behalf of the Father and on behalf of Christ.
By carefully observing such expressions we may understand more of the revelation of God by his own word.