St. Alphonsus has a great article on the subject:
https://sensusfidelium.us/apologetics/history-of-heresies-their-refutation-st-alphonsus/the-errors-of-the-greeks-condemned-in-three-general-councils/St. Robert Bellarmine wrote another that is online here:
http://www.catholicapologetics.info/apologetics/protestantism/procession.htmRecall that St. Athanasius had already taught us in his Creed, even before Constantinople I Council in 381 A.D., "The Holy Ghost is of the Father and of the Son; not made nor created nor begotten, but proceeding". St. Athanasius also taught elsewhere "Jointly with the Father, the Son is indeed the Fount of the Holy Spirit". St. Ambrose, St. Augustine, Popes St. Damasus, St. Leo etc taught the Filioque.
The two internal processions in the Holy Trinity are (1) The eternal generation of the Son, and (2) the co-eternal procession of the Holy Spirit from the Father through the Son. This is the way both St. Augustine and St. Cyril explain it. As the Father eternally gave birth to the Son, so likewise the Father eternally gave to the Son, that the Holy Spirit should proceed eternally from Him.
Already, at the Council of Toledo in 589 A.D., Archbishop St. Leander of Seville required the former Arian king to profess the Filioque to be reconciled back to the Church. Archbishop St. Isidore of Seville, his brother, who knew all of Hebrew, Greek and Latin, re-affirmed the Filioque saying, "There is this difference between the Generation of the Son and the Procession of the Spirit; the Son is begotten of One, while the Spirit proceeds from Both". St. Fulgentius of Ruspe says in Rule of Faith, "Hold most firmly and never doubt in the least that the same Holy Spirit, the One Spirit of the Father and of the Son, proceeds from the Father and the Son. That He proceeds also from the Son is supported by the testimony of both Prophets and Apostles". As St. Robert writes, Pope St. Leo had said, "Ninth Blessed Leo, “There is one who generates; another who is generated; another who proceeds from both.” And this is that Leo the Great whom in the fourth Synod 630 Bishops, almost all Oriental, extolled with the greatest praise, and about whom they repeated again and again that as Leo believes so also do we believe." St. Robert proves the Dogma from 15 Greek Fathers, 15 Latin Fathers and 5 General Councils.
At the Seventh Ecuмenical Council, Patriarch St. Tarasius of Constantinople, of the Greek Church, professed in the Greek tongue, "το Πνευμα το αγιον, το κυριον και ζωοποιον, το εκ
του Πατροσ δια του Υιου εκπορευομενον (ekporeuomenon)". The meaning is, "And in the Holy Spirit, the Lord and Life-Giver, Who Proceeds from the Father through the Son". It is true that ekporeuomenon or ekporeusthai in Greek has a slight difference in meaning from our procedere or procedit in Latin, however, eternal procession is conveyed in both terms.
In Rev 22
http://www.newadvent.org/bible/rev022.htm the Living Water that proceeds from "God and the Lamb" (from the Father and the Son) is the Holy Spirit, according to St. Ambrose and other Fathers. If you see the concordance there, it is procedentem in Latin and ἐκπορευόμενον (ekporoumenon) in Greek. St. Augustine also says Jesus breathed the Spirit on His Apostles to show the Holy Spirit proceeded from Him, just as He proceeded from God the Father. This is what St. Cyril says also in that Ecuмenical Letter.
But Photius later claimed that the Spirit was given to the Son only temporally, not eternally, and this heresy is crypto-Arianism; for if the Son did not eternally have the Holy Spirit, it would follow, that He is not equal to the Father, and is not eternally God. Our Lord said that He has all that the Father has; therefore, the spiration of the Holy Spirit belongs to the Son just as it belongs to the eternal Father.
The Greek Church was given two opportunities, at the Council of Lyons II and Florence, as St. Alphonsus docuмents in detail, to come back to Holy Union with the Catholic Church. Happily some 23 odd Eastern Catholic Churches remain in full communion with the Catholic Church; but unfortunately, many of the schismatic sects remain separated. The future resolution must involve some kind of joint Council imho, where the Greek Orthodox retract their errors and submit to the Church. There is a prophesy of a future Ecuмenical Council at which "The Greeks will return to the obedience of the Holy Roman Church." After the schism, they fell into other heresies, for e.g. some denied Purgatory, the Immaculate Conception, etc. It almost always happens, as St. Jerome observes, that schism ends in heresy also.