Does the Holy Spirit's procession from the Father and Son infringe on the co-equality of the Trinity? The following is intended to correlate to classical scholastic terminology, to whit:

Nature = essence, as in "the Father, Son, and Holy Spirit all have the Trinitarian 'nature' of God."

Person = mode of particular operation, as in "the Son is the second "person" of the Trinity, light from light, true God from true God, begotten not made..."

The Father, Son, and Holy Spirit are all co-eternal and co-equal.

Question

If the the persons of the Trinity - Father, Son, and Holy Spirit - are co-eternal and co-equal, and the Holy Spirit proceeds from both the Father and the Son...

Wouldn't the Father and the Son need to proceed from the Spirit as well in order to maintain the triad of equal procession from all three?

It seems as though, since the Holy Spirit doesn't have the faculty of bestowing procession upon the Father and the Son, then that gives the Holy Spirit one less faculty than the rest, which in turn subordinates the Holy Spirit to receive procession only.  

Doesn't this infringe on the Spirit's co-equality?

This question derives from my study of the and drawn out debate concerning the validity of the filioque.
Brief Historical Introduction

The Nicene Creed originally did not include the words "and the Son" (called the Filioque clause) because it was based on the words of Scripture in John 15:26 (τὸ ἐκ τοῦ Πατρὸς ἐκπορευόμενον). It was added later, and not by an ecumenical council (it was originally inserted by the Third Synod of Toledo). Not to mention, even previous Roman popes resisted its addition to the Creed (e.g. Leo III). It adds nothing to the Creed that isn't already said elsewhere in it. Writing in the fourth century, Gregory of Nazianzus wrote concerning the Holy Spirit:


  The Holy Ghost is truly Spirit, coming forth from the Father indeed,
  but not after the manner of the Son, for it is not by Generation but
  by Procession (since I must coin a word for the sake of clearness);
  for neither did the Father cease to be Unbegotten because of His
  begetting something, nor the Son to be begotten because He is of the
  Unbegotten (how could that be?), nor is the Spirit changed into Father
  or Son because He proceeds, or because He is God— though the ungodly
  do not believe it (Oration 39:12).


The Greek Church Fathers were careful not to use the verb ἐκπορεύομαι to describe the relation of the Holy Spirit from the Son, this term was exclusively for how the Holy Spirit proceeds from the Father (and is directly from the mouth of Jesus as recorded in John's Gospel).1 

In contrast, Western Christians had begun using the Latin verb procedere to translate ἐκπορεύομαι and to describe the Holy Spirit's relationship to both the Father and the Son. History shows that some Christians were indifferent to this and others took issue with it. Controversies ensued that I won't go into as you can read the history in more detail elsewhere. The Roman Catholic Church has even officially admitted that the Holy Spirit does not proceed (ἐκπορεύομαι) from the Son, but rather from the Father through the Son, however it maintains the current verbiage of the creed with caveats to explain this distinction (with the exception of Eastern Catholic churches, which it encourages to omit the Filioque).

Does the Holy Spirit's procession from the Father and Son infringe on the co-equality of the Trinity?

Yes. John 15:26 says that the Holy Spirit proceeds from the Father (ὃ παρὰ τοῦ πατρὸς ἐκπορεύεται / who from the Father proceeds). That the Holy Spirit proceeds from the Father is straight from Jesus' mouth as recorded in Scripture. Jesus didn't say, "and from me." The Filioque makes the Holy Spirit a subordinate member of the Trinity. The procession of the Holy Spirit is now no longer unique to only one Person of the Trinity nor shared by all Persons, thus compromising the co-equality of the Persons of the Trinity.



1 This is evidenced by the Alexandrian fathers' use of προϊέναι and προχεῖσθαι rather than ἐκπορεύεσθαι, as an example.