St. Robert and St. Alphonsus are among the foremost experts on the Papacy and on Church History. Go with them, and you'll never fall into serious error, especially when they both clearly tell you the case of Pope Honorius has been abused by enemies of the Roman Church (such as the Greek Orthodox, also the Gallicans whom the Doctors controvert, and whose opinion Vatican I finally dogmatically rejected; it is a Greek Orthodox calumny and a Gallican slander that many Roman Pontiffs have allegedly been heretics, which modern neo-Gallicans have cheerfully adopted as their own calumny). No, the Roman Pontiffs, the Vicars of Christ, the Successors of St. Peter, can never become heretics, nor fail in the Faith, for the Lord has prayed and the Scripture has borne witness, that the Faith of Peter will never fail. Pope Honorius made a mistake, for he should have condemned the heresy and uprooted it. Two saints of the day were Pope St. Martin and St. Maximus of Constantinople, the Byzantine monk who was the Athanasius of the age against Monothelitism; the heretical Patriarch Phyrrus had tried to convince him, and been defeated. Thanks to him and to Pope St. Martin, a first victory over the heresy was won. St. Maximus had his tongue cut out by the imperial forces and was martyred.
1. St. Maximus: "they have not conformed to the sense of the Apostolic see, and what is laughable, or rather lamentable, as proving their ignorance, they have not hesitated to lie against the Apostolic see itself . . . but have claimed the great Honorius on their side. . . . What did the divine Honorius do, and after him the aged Severinus, and John who followed him? Yet further, what supplication has the blessed pope, who now sits, not made? Have not the whole East and West brought their tears, laments, obsecrations, deprecations, both before God in prayer and before men in their letters? If the Roman see recognizes Pyrrhus to be not only a reprobate but a heretic, it is certainly plain that everyone who anathematizes those who have rejected Pyrrhus, anathematizes the see of Rome that is, he anathematizes the Catholic Church. I need hardly add that he excommunicates himself also, if indeed he be in communion with the Roman see and the Church of God.... It is not right that one who has been condemned and cast out by the Apostolic see of the city of Rome for his wrong opinions should be named with any kind of honour, until he be received by her, having returned to her — nay, to our Lord — by a pious confession and orthodox faith, by which he can receive holiness and the title of holy.... Let him hasten before all things to satisfy the Roman see, for if it is satisfied all will agree in calling him pious and orthodox. For he only speaks in vain who thinks he ought to persuade or entrap persons like myself, and does not satisfy and implore the blessed pope of the most holy Church of the Romans, that is, the Apostolic see, which from the incarnate Son of God Himself, and also by all holy synods, according to the holy canons and definitions, has received universal and supreme dominion, authority and power of binding and loosing over all the holy Churches of God which are in the whole world — for with it the Word who is above the celestial powers binds and looses in heaven also. For if he thinks he must satisfy others, and fails to implore the most blessed Roman pope, he is acting like a man who, when accused of murder or some other crime, does not hasten to prove his innocence to the judge appointed by the law, but only uselessly and without profit does his best to demonstrate his innocence to private individuals, who have no power to acquit him."
http://www.newadvent.org/cathen/10078b.htmFr. Butler's entry on Patriarch St. Sophronius of Jerusalem, who lived during this time, has more on the case of Pope Honorius, "He was no sooner established in his see, than he assembled a council of all the bishops of his patriarchate, in 634, to condemn the Monothelite heresy, and composed a synodal letter to explain and prove the Catholic faith. 'Finis excellent piece was confirmed in the sixth general council. St. Sophronius sent this learned epistle to pope Honorius and to Sergius. This latter had, by a crafty letter and captious expressions, persuaded pope Honorius to tolerate a silence as to one or two wills in Christ. It is evident from the most authentic monuments, that Honorius never assented to that error, but always adhered to the truth.[1] However, a silence was ill-timed, and though not so designed, might be deemed by some a kind of connivance, for a rising heresy seeks to carry on its work under ground without noise: it is a fire which spreads itself under cover. Sophronius, seeing the emperor and almost all the chief prelates of the East conspire against the truth, thought it his duty to defend it with the greater zeal. He took Stephen, bishop of Doria, the eldest of his suffragans, led him to Mount Calvary, and there adjured him by Him who was crucified on that place, and by the account which he should give him at the last day, "to go to the apostolic see, where are the foundations of the holy doctrine, and not to cease to pray till the holy persons there should examine and condemn the novelty." Stephen did so and stayed at Rome ten years, till he saw it condemned by pope Martin I. in the council of Lateran, in 649"
2. The dogmatic letter of Pope St. Agatho which was read at the Sixth Council, after which the Council Fathers said "Peter has spoken through Agatho" states, "For if anybody should mean a personal will, when in the holy Trinity there are said to be three Persons, it would be necessary that there should be asserted three personal wills, and three personal operations (which is absurd and truly profane). Since, as the truth of the Christian faith holds, the will is natural, where the one nature of the holy and inseparable Trinity is spoken of, it must be consistently understood that there is one natural will, and one natural operation. But when in truth we confess that in the one person of our Lord Jesus Christ the mediator between God and men, there are two natures (that is to say the divine and the human), even after his admirable union, just as we canonically confess the two natures of one and the same person, so too we confess his two natural wills and two natural operations ... because the true confession thereof for which Peter was pronounced blessed by the Lord of all things, was revealed by the Father of heaven, for he received from the Redeemer of all himself, by three commendations, the duty of feeding the spiritual sheep of the Church; under whose protecting shield, this Apostolic Church of his has never turned away from the path of truth in any direction of error, whose authority, as that of the Prince of all the Apostles, the whole Catholic Church, and the Ecuмenical Synods have faithfully embraced, and followed in all things; and all the venerable Fathers have embraced its Apostolic doctrine, through which they as the most approved luminaries of the Church of Christ have shone; and the holy orthodox doctors have venerated and followed it, while the heretics have pursued it with false criminations and with derogatory hatred. This is the living Tradition of the Apostles of Christ, which his Church holds everywhere, which is chiefly to be loved and fostered, and is to be preached with confidence ... For this is the rule of the true faith, which this spiritual mother of your most tranquil empire, the Apostolic Church of Christ, has both in prosperity and in adversity always held and defended with energy; which, it will be proved, by the grace of Almighty God, has never erred from the path of the apostolic tradition, nor has she been depraved by yielding to heretical innovations, but from the beginning she has received the Christian faith from her founders, the princes of the Apostles of Christ, and remains undefiled unto the end, according to the divine promise of the Lord and Saviour himself, which he uttered in the holy Gospels to the prince of his disciples: saying, Peter, Peter, behold, Satan has desired to have you, that he might sift you as wheat; but I have prayed for you, that (your) faith fail not. And when you are converted, strengthen your brethren. Let your tranquil Clemency therefore consider, since it is the Lord and Saviour of all, whose faith it is, that promised that Peter's faith should not fail and exhorted him to strengthen his brethren, how it is known to all that the Apostolic pontiffs, the predecessors of my littleness, have always confidently done this very thing"
http://www.newadvent.org/fathers/3813.htmThis is a dogmatic letter and it confirms Pope St. Agatho and the Sixth Council judged the Apostolic Church of Rome had never deviated from the path of Truth.
A Council must submit all its acts for confirmation to the Pope, as for e.g. Patriarch St. Anatolius of Constantinople did to Pope St. Leo I. The Roman Pontiffs (who were likely misled by forged acts anyway, as the Doctors tell us) did not accept it in the wrong sense of the Greeks and the Gallicans, but only in a much more limited manner, namely that Pope Honorius should have done his duty better. And it has been reversed now, in light of new evidence, and Pope St. Agatho's letter is right.