The Council of Ephesus in 431 AD teaches with St. Cyril of Alexandria, that the Word Himself stemming from God became our High Priest, when He became man and flesh like us.
Thus as man He is a priest.
In the Old Covenant Christ's priesthood is foretold in Ps.109:4 :
" The Lord hath sworn and he will not repent: Thou art a priest for ever according to the Order of Melchisedech."
The Messianic character of these words is confirmed in Holy Scripture
Mt. 22, 42; Hebr. 5, 6, 10; 7, 17, 21.
The Epistle to the Hebrews contains a formal treatise on Christ's priesthood . The author points out that Christ personally fulfilled all demands of the priesthood.
" For every high priest taken from among men is ordained for men in the things that appertain to God. that he may offer up gifts and sacrifices for sins; Christ had human nature in common with us men so that he might have sympathy with our weaknesses. Heb 5:1
He was called by God to the priesthood ; for all those who obey Him He is the author of eternal salvation by offering Himself on the Cross as a sacrifice of expiation.
Christ's priesthood is exalted over the Levitic priesthood of the Old Covenant. Compare the relation of Melchisedech who was a model of Christ, to Abraham. According to Ps. 109, 4, Christ was installed in the priestly office through an oath of God; He possesses a priesthood which does not pass;
He is holy, innocent, immaculate, separated from all sinners;
He is the Son of God consummated in eternity;
through His unique act of self-sacrifice
He washed away the sins of men.
The Fathers, from the very beginning, reiterate the thoughts of the Epistle to the Hebrews. St. Clement of Rome calls Christ: " the High Priest of our sacrificial gifts". St. Ignatius of Antioch says in regard to Christ's priesthood : " Good are the priests (of the Old Covenant), but better is the High Priest to Whom the All-Highest is entrusted ". St. Polycarp calls Jesus Christ,'. The Eternal High Priest " .
Christ's priesthood begins with the Hypostatic Union.
The proper task of the priest consists in being a mediator between God and men Christ's priesthood is of eternal duration, as the sacerdotal dignity of Christ founded in the Hypostatic Union remains for ever, and also because the operation of His priesthood endures eternally in those redeemed who enjoy for ever the Beatific Vision.
Again, Christ's sacrificial disposition, as far as it consists in praise and thanksgiving, endures for eternity.
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The Exercise of the Sacerdotal Office or Christ's Sacrifice
Concept of Sacrifice
The most essential function of the sacerdotal office is sacrifice. Hebr. 8, 3
For every high priest is appointed to offer gifts and sacrifices."
The Council of Trent declares ..Sacrifice and priesthood are, through God's ordinance, so connected with each other that both existed in every order of salvation.
By sacrifice is understood in the widest sense, the surrender of some good for the sake of a good aim. The religious meaning attaching to sacrifice in the wider sense is every inner act of self-surrender to God, and every outer manifestation of the inner sacrificial disposition, that is, prayer, almsgiving, mortification.
In the narrower liturgical sense one takes sacrifice to mean an external religious act, in which a gift perceptible to the senses is offered by an ordained servant of God in recognition of the absolute sovereignty and majesty of God, and, since the Fall, in atonement to God.
In a sacrifice we distinguish :
a) A visible sacrificial gift which represents the thing being sacrificed,
b) A sacrificing priest who is authorized to appear before God as the representative of the community,
c) The purpose of the sacrifice, which primarily consists in the recognition of the absolute majesty of God through adoration, thanksgiving, entreaties, and secondarily in the reconciliation with God through atonement,
d) An act of sacrifice which represents in a way apparent to the senses the inner sacrificial disposition through the offering of the sacrificial gift.
Excerpted in part: Fundamentals of Catholic Dogma, Ludwig Ott, pp182-3