Correct.
The Godhead of the Father, the Son, and the Holy Spirit refers to the Divine Essence, Being, Substance, and Nature. So the Godhead of the Father is the Godhead of the Son is the Godhead of the Holy Spirit; for the Three are One in Godhead--i.e. the Three Persons are each God, the one and only and undivided God.
From the 11th Council of Toledo (675 AD):
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We confess and believe the holy and ineffable Trinity, the Father, and the Son, and the Holy Spirit, one God naturally, to be of one substance, one nature, and also of one majesty and power. And we profess that the Father, indeed, is not begotten, not created but unbegotten. For He from whom both the Son received His birth and the Holy Spirit His procession takes His origin from no one. Therefore, He is the source and origin of all Godhead [i.e. Divinity]
; also is the Father Himself of His own essence, He who ineffably begot the Son from an ineffable substance; nor did He, however, beget other than what He Himself is: God God, light light," -
Source
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That he is Father we know in utmost fullness from Jesus Christ, who constantly makes loving, thankful, and reverent reference to him as his Origin. It is because he bears fruit out of himself and requires no fructifying that he is called Father, and not in the sexual sense, for he will be the Creator of man and woman, and thus contains the primal qualities of woman in himself in the same simultaneously transcending way as those of man. (The Greek gennad can imply both siring and bearing, as can the word for to come into being: ginomai.) Jesus’ words indicate that this fruitful self-surrender by the primal Origin has neither beginning nor end: It is a perpetual occurrence in which essence and activity coincide. Herein lies the most unfathomable aspect of the Mystery of God: that what is absolutely primal is no statically self-contained and comprehensible reality, but one that exists solely in dispensing itself: a flowing wellspring with no holding-trough beneath it, an act of procreation with no seminal vesicle, with no organism at all to perform the act. In the pure act of self-pouring-forth, God the Father is his self, or, if one wishes, a “person” (in a transcending way)." - Fr. Hans Urs von Balthasar, Credo
The monarchy of the Father means that the the Father is the "Fount of Deity", for He receives His Godhead (Divinity, Deity) from none. The Son and Spirit are God because they are homoousios with the Father; thus the Son is God because the Father is God, and the Spirit is God because the Father is God. Thus the eternal generation of the Son means He is God of God, Light of Light, true God of true God. The same is likewise true of the Spirit by His eternal procession from the Father.
The Son is God because the Father is God, and the Son has His eternal generation from the Father.
The Spirit is God because the Father and Son are God, and the Spirit has His eternal procession from the Father [and the Son] (if we follow the Filioque here).
However, the Father is never the Father by Himself, He is always with the Son and the Spirit. There is never a time when the Son was not, never a time when the Spirit was not. The Father is not the Father except that He is Father of the Son obviously. So none of this can be any kind of heretical Subordinatism, with the Son and Spirit lesser than the Father.
We are not speaking in such a way; but rather are speaking of the inter-personal relationality of the Three in their Eternal Perichoresis.
-CryptoLutheran