Where does the belief come from? Why is her intercession believed to be necessary? Please provide Biblical evidence for the belief as well as Traditional sources.
Mary is co-redemptrix just means she has cooperated with Jesus Christ for our salvation. Explanations can be found in Lumen gentium quoted below :
There is but one Mediator as we know from the words of the apostle, "for there is one God and one mediator of God and men, the man Christ Jesus, who gave himself a redemption for all" (1Tm 2,5-6). The maternal duty of Mary toward men in no wise obscures or diminishes this unique mediation of Christ, but rather shows His power.
Thus Mary, a daughter of Adam, consenting to the divine Word, became the mother of Jesus, the one and only Mediator. Embracing God's salvific will with a full heart and impeded by no sin, she devoted herself totally as a handmaid of the Lord to the person and work of her Son, under Him and with Him, by the grace of almighty God, serving the mystery of redemption. Rightly therefore the holy Fathers see her as used by God not merely in a passive way, but as freely cooperating in the work of human salvation through faith and obedience.
This union of the Mother with the Son in the work of salvation is made manifest from the time of Christ's virginal conception up to His death it is shown first of all when Mary, arising in haste to go to visit Elizabeth, is greeted by her as blessed because of her belief in the promise of salvation and the precursor leaped with joy in the womb of his mother (cf. Lc 1,41-45). This union is manifest also at the birth of Our Lord, who did not diminish His mother's virginal integrity but sanctified it, (Cf Eph. 1:18) when the Mother of God joyfully showed her firstborn Son to the shepherds and Magi. When she presented Him to the Lord in the temple, making the offering of the poor, she heard Simeon foretelling at the same time that her Son would be a sign of contradiction and that a sword would pierce the mother's soul, that out of many hearts thoughts might be revealed (cf. Lc 2,34-35). When the Child Jesus was lost and they had sought Him sorrowing, His parents found Him in the temple, taken up with the things that were His Father's business; and they did not understand the word of their Son. His Mother indeed kept these things to be pondered over in her heart (cf. Lc 2,41-51).
In the public life of Jesus, Mary makes significant appearances. This is so even at the very beginning, when at the marriage feast of Cana, moved with pity, she brought about by her intercession the beginning of miracles of Jesus the Messiah (cf. Jn 2,1-11). In the course of her Son's preaching she received the words whereby in extolling a kingdom beyond the calculations and bonds of flesh and blood, He declared blessed (cf. Mc 3,35 par. and Lc 11,27-28) those who heard and kept the word of God, as she was faithfully doing (cf. Lc 2,19 2,51). After this manner the Blessed Virgin advanced in her pilgrimage of faith, and faithfully persevered in her union with her Son unto the cross, where she stood, in keeping with the divine plan, (Cf. Jn. 19:25) grieving exceedingly with her only begotten Son, uniting herself with a maternal heart with His sacrifice, and lovingly consenting to the immolation of this Victim which she herself had brought forth. Finally, she was given by the same Christ Jesus dying on the cross as a mother to His disciple with these words: "Woman, behold thy son" (cf. Jn 19,26-27).
Why do many Catholics believe that Mary is co-redemptrix?
The short answer is: Because it is not yet defined and as such are permitted to do so. Some Catholics do not believe this a dogmatic revealed truth, yet some do believe in this concept of undefined dogma.
Undefined dogmas leaves Catholics free to believe this subject or put it aside. Wait 50 years when Rome defines this question and the Papal Bull will give you all the Biblical references and a list of traditions that the Church uses in its definition of the Co-Redemptrix of Mary. Or maybe Rome will eventually define the opposite is true.
Nevertheless there seems to be some Scriptural basis for this.
The New Testament is commonly cited in favour of this teaching:
John 19:25-27: "Now there stood by the cross of Jesus, his mother, and his mother's sister, Mary of Cleophas, and Mary Magdalen. When Jesus therefore had seen his mother and the disciple standing whom he loved, he saith to his mother: Woman, behold thy son. After that, he saith to the disciple: Behold thy mother. And from that hour, the disciple took her to his own." Lumen gentium, the Dogmatic Constitution on the Church, states, "...In this singular way she cooperated by her obedience, faith, hope and burning charity in the work of the Saviour in giving back supernatural life to souls. Wherefore she is our mother in the order of grace."
Colossians 1:24: "[I, Paul,] now rejoice in my sufferings for you, and fill up those things that are wanting of the sufferings of Christ, in my flesh, for his body, which is the church”. If Paul could fill up what was behind, so could the Blessed Virgin Mary a fortiori. At the same time, this is easily interpreted as the task of all Christians to mediate the face of Christ to the world.
Proposed dogmatic definition
There have been efforts to propose a formal dogmatization, which has had both popular and ecclesiastical support. The proposal for the dogma is often associated with the alleged apparitions of The Lady of All Nations to Ida Peerdeman, in Amsterdam, Netherlands. The visionary reported that the Lady of the apparition repeatedly instructed her to petition Pope Pius XII to dogmatically define Mary's spiritual motherhood under the threefold title of Coredemptrix, Mediatrix, and Advocate. In the intervening decades, the Church's stance on the Amsterdam apparitions has been mixed. As of 2002, the apparitions have the approval of the diocesan ordinary, Bishop Jozef Marianus Punt of Haarlem-Amsterdam.
The possibility of such a dogma was brought up at the Second Vatican Council by Italian, Spanish, and Polish bishops, but not dealt with on the council floor. Subsequently, "not only did the Council not take the route of a dogmatic pronouncement, but it positively avoided using 'coredemptio'", and popes pointedly did not include such language in their encyclicals.
In the early 1990s Prof. Mark Miravalle of the Franciscan University of Steubenville and author of the book Mary: Coredemptrix, Mediatrix, Advocate launched a popular petition to urge Pope John Paul II to declare Mary Coredemptrix ex cathedra.
Salvatore Perrella, O.S.M., of the Pontifical Theological Faculty of the Marianum in Rome, thought that this indicated "...a certain 'under-appreciation' of the Council's teaching, which is perhaps believed to be not completely adequate to illustrate comprehensively Mary's co-operation in Christ's work of Redemption."
Arguments opposed are that such a dogma might limit, in popular understanding, the redemptive role of Jesus Christ. Faber says,
Our Blessed Lord is the sole Redeemer of the world in the true and proper sense of the word, and in this sense no creature whatsoever shares the honor with Him, neither can it be said of Him without impiety that He is co-redeemer with Mary, ... [although] in a degree to which no others approach, our Blessed Lady co-operated with Him in the redemption of the world.
Faber recognized that the term Co-Redemptrix usually requires some explanation in modern English because so often the prefix co- tends to imply complete equality. He also explains that, "Thus, so far as the literal meaning of the word is concerned, it would appear that the term co-redemptress is not theologically true, or at least does not express the truth it certainly contains with theological accuracy."
This concern is shared by Perrella.
The semantic weight of this expression would require a good many other qualifications and clarifications, especially in the case under examination, where she who is wished to be proclaimed co-redeemer is, in the first place, one who is redeemed, albeit in a singular manner, and who participates in Redemption primarily as something she herself receives. Thus we see the inadequacy of the above-mentioned term for expressing a doctrine which requires, even from the lexical standpoint, the proper nuances and distinctions of levels.
It was rejected by the Vatican in the past because of serious theological difficulties. In August 1996, a Mariological Congress was held in Częstochowa, Poland, where a commission was established in response to a request of the Holy See. **The congress sought the opinion of scholars present there regarding the possibility of proposing a fifth Marian dogma on Mary as Coredemptrix, Mediatrix, and Advocate. The commission unanimously declared that it was not opportune, voting 23-0 against the proposed dogma.
Another argument is that it would also complicate ecumenical efforts for a better understanding of the role of the Blessed Virgin Mary in the salvation mystery of Jesus Christ.
By 1998 it was doubtful the Vatican was going to consider new Marian dogmas. The papal spokesman stated "This is not under study by the Holy Father nor by any Vatican congregation or commission." A leading Mariologist stated the petition was "theologically inadequate, historically a mistake, pastorally imprudent and ecumenically unacceptable." Pope John Paul II cautioned against "all false exaggeration"; his teaching and devotion to Mary has strictly been "exalting Mary as the first among believers but concentrating all faith on the Triune God and giving primacy to Christ." In his 1994 Apostolic letter, Tertio Milennio Adveniente, John Paul said, "Christ, the Redeemer of the world, is the one Mediator between God and men, and there is no other name under heaven by which we can be saved (cf. Acts 4:12)." When asked in an interview in 2000 whether the Church would go along with the desire to solemnly define Mary as Co-redemptrix, then-Cardinal Ratzinger (later Pope Benedict XVI) responded that,
the formula “Co-redemptrix” departs to too great an extent from the language of Scripture and of the Fathers and therefore gives rise to misunderstandings. ...Everything comes from Him [Christ], as the Letter to the Ephesians and the Letter to the Colossians, in particular, tell us; Mary, too, is everything she is through Him. The word “Co-redemptrix” would obscure this origin. A correct intention being expressed in the wrong way.
Benedict further explained his notable opposition of a dogmatization, concluding that the title is sufficiently included in other better expressions of Catholic Marian teaching. "For example, the Scriptural account is unsatisfactory, and above all, we are talking most of the time of a merit de congruo which would seem, by the very definition of de congruo, not fit into the exact clearness needed for dogmatic definitions."
René Laurentin, theologian specializing in Mariology, said "“There is no mediation or co-redemption except in Christ. He alone is God.” - Co-Redemptrix
At least it has not been officially promulgated by any pope so far.
However, Pope Francis is definitely not in favour of declaring this a dogma of the Church.
Pope calls idea of declaring Mary co-redemptrix ‘foolishness’
Pope Francis appeared to flatly reject proposals in some theological circles to add “co-redemptrix” to the list of titles of the Virgin Mary, saying the mother of Jesus never took anything that belonged to her son, and calling the invention of new titles and dogmas “foolishness.”
“She never wanted for herself something that was of her son,” Francis said. “She never introduced herself as co-redemptrix. No. Disciple,” he said, meaning that Mary saw herself as a disciple of Jesus.
Mary, the pope insisted, “never stole for herself anything that was of her son,” instead “serving him. Because she is mother. She gives life.”
“When they come to us with the story of declaring her this or making that dogma, let’s not get lost in foolishness [in Spanish, tonteras],” he said.
Francis’ words, delivered in Spanish, came while celebrating a Thursday evening Mass in Rome for the Feast of Our Lady of Guadalupe.
The title of Mary as “co-redemptrix” dates to the Middle Ages, and the idea of declaring it as a church dogma was discussed, though not adopted, at the Second Vatican Council. In the 1990s American Catholic theologian Mark Miravalle launched a petition asking the pope to make such a declaration, and today the “co-redemptrix” devotion tends to be strongest among more conservative Catholics.
What Francis said Thursday is in line with Cardinal Joseph Ratzinger, the Vatican’s doctrinal chief during most of St. John Paul II’ papacy, and now Pope emeritus Benedict XVI.
Speaking with Peter Seewald for the book-length interview published as God and the World: A Conversation, the then cardinal said: “The formula ‘co-redemptrix’ departs to too great an extent from the language of Scripture and of the Fathers, and therefore gives rise to misunderstandings.” “Everything comes from Him [Christ], as the Letter to the Ephesians and the Letter to the Colossians, in particular, tell us; Mary, too, is everything she is through Him,” Ratzinger said. “The word ‘co-redemptrix’ would obscure this origin. A correct intention being expressed in the wrong way.”
In his other remarks, Francis said the celebration held in St. Peter’s basilica, the passage of the Gospel read and the print of la Morenita that stood tall in front and to the side of the altar made him think of three adjectives for Mary: “woman-Lady-, mother and mestiza [mixed race].”
“Mary introduces herself as a woman,” he said. “And she introduces herself with the message of someone else. Meaning, she’s woman and she’s disciple.” “Christian piety, throughout the ages, always sought to praise her with new titles,” Francis said. “They were filial titles,” that expressed “the love of the people of God, but they did not touch in any way her being a woman disciple.” - Pope calls idea of declaring Mary co-redemptrix ‘foolishness’
Why many Catholics believe that Mary is co-redemptrix? (New Question)
The short answer is, because Mary Co-Redemptrix was part of the Church Tradition since the time of St.Irenaeus in 200AD, and since he is a Church Father he is close to the Apostolic Traditions and understood the teachings of the Apostles about the role of Mary.
But the most important thing is the word of Jesus who acknowledged Mary's role as Co-Redemptrix being the source of His Redeeming Sacrifice who gave Her body as "Co-Redemptrix" and was reflected upon by this article link saying;
Saint John Damascene says that the Lord Himself gave Her last communion to Her saying, “Receive, my Lady and Mother, from my own hands the Body which You gave Me and which was formed in Your precious womb.” https://www.homeofthemother.org/en/resources/virgin-mary/meditations-on-mary/8304-mary-s-death
IMPORTANT NOTE: The word "Co-redemptrix" may not appear in scriptures but like the word "Trinity" both did not appear but it doesn't mean that the scriptures does not teaches it, it takes pondering to discover & understand fully God given role to Mary.
What is the basis for the Catholic belief that Mary is co-redemptrix?(Old Question)
The following link provides important doctrines why the title of Mary as Co-redemptrix is rightfully fitting title for Our Lady citing St. John Paul II prayers, homilies and public addressed:
POPE JOHN PAUL II'S ORDINARY MAGISTERIUM ON MARIAN COREDEMPTION: CONSISTENT TEACHING AND MORE RECENT PERSPECTIVES by Monsignor Arthur Burton Calkins
But in a more simple approach, St. Pius X in 1906 provided a very clear biblical foundation in the gospel and concluded by a well known early Church Father St.Irenaeus.
Jesus is the Redeemer because He is the High Priest that offer a pleasing sacrifice to God in atonement for all the sins of mankind. What did Jesus offered? Jesus offered His sinless "body & blood".
Is Mary a Co-redeemer or Co-redemptrix? Yes, how?
Mary like Jesus is also a Priest like Jesus. In 1906, St. Pius X approved the prayer devotion to Mary as Virgin Priest and granting a 300 days indulgences to all the faithful who implore Mary's intercession by invoking the prayer "Mary Virgin Priest, pray for us."
If Mary was acknowledged by the Church as a Virgin Priest, did She offered a pleasing "sacrifice" too like Jesus the High Priest? Yes, how?
Jesus in the Last Supper had teach the Apostles how to offer a "sacramental mass" by saying this words "this is my body, this is my blood". The consecration prayer was pronounced and then the Holy Spirit come down to sanctifies the "bread & wine" and form Jesus Christ. How about Mary, the Virgin Priest, did She pronounce a "sacramental offering" like the Apostles? No!
The Mystery of Annunciation was acknowledged by the theologians in the 17th century (see link below) as Mary's "ordination" into a royal priesthood and St. Pius X proclaimed Mary as Virgin Priest. The words "Hail, Full of Grace, the Lord is with you." is not just a mere greetings or salutation but a formula for ordination into priesthood like what a Bishop or Prophets do in ordaining Priest or Bishops, this is a profound mystery.
In the Mystery of Incarnation, Mary said Her Fiat, this is the First Holy Mass of the Blessed Virgin Mary as Priest, by saying the in effect "this is my body, this is my blood" and *"Let it be done to me according to your Word". After Mary's Fiat or consecration prayer offering Her "sinless body & blood", the Holy Spirit came down and fecundated Her pure womb and formed Jesus Christ, like what Priest is doing making Jesus present in the Holy Eucharist. Mary's Fiat transformed Her sinless "body & blood" like a Living Host, the Tabernacle of God and the New Ark of the Covenant. Since, Mary transformed into a Living Host the Holy Spirit must "overshadowed" Her like the Ark of the Covenant or the Church Tabernacle must be veiled upon.
Mary is Co-redemptrix because She is a Virgin Priest that offered Her "sinless body & blood" to God as a pleasing sacrifice to God for the Redeemer Jesus Christ to use for the atonement of the sins of mankind.
- St. Ireneaus had said "Mary is the cause of our salvation", because without Mary's Fiat or Her consecration prayer offering Her sinless body & blood, Jesus the Redeemer has no way to redeem mankind, because the Logos is a Spirit needed the cooperation of the chosen Woman in eternity destined to be the Theotokos.
In closing, Mary is Co-Redemptrix is the Will of God written in His Holy Decree after the fall of Adam & Eve.
Genesis 3:15 Douay-Rheims 1899 American Edition (DRA) 15 I will put enmities between thee and the woman, and thy seed and her seed: she shall crush thy head, and thou shalt lie in wait for her heel.
Simple analogy over the role of Redeemer vs. Co-Redemptrix;
St.Paul teaches in the his letter to the Romans;
A Living Sacrifice 12 Therefore, I urge you, brothers and sisters, in view of God’s mercy, to offer your bodies as a living sacrifice, holy and pleasing to God—this is your true and proper worship. 2 Do not conform to the pattern of this world, but be transformed by the renewing of your mind. Then you will be able to test and approve what God’s will is—his good, pleasing and perfect will. Romans 12 New International Version (NIV)
Jesus the High Priest offered his "sinless body & blood" as a pleasing sacrifice to God in atonement for the sins of mankind and become the Redeemer of Mankind.
Mary is a Virgin Priest who offered Her "sinless body & blood" thru Her Fiat as a pleasing sacrifice to God for all mankind to be save by Jesus Christ the Redeemer and became the Co-redemptrix.
How about, the Apostles, Church Fathers, Saints, Martyrs, Bishops, Priest and ordinary laity? Are we "co-redeemers" too? Yes, even the ordinary laity upon baptism was ordained into common priesthood and can offer their bodies as a living sacrifice as St.Paul described in Romans12:1 and explain in Collosians1:4
"Now I rejoice in my sufferings for your sake, and in my flesh I do my share on behalf of His body, which is the church, in filling up what is lacking in Christ's afflictions.(Collosians1:24)
"As you come to him, the living Stone—rejected by humans but chosen by God and precious to him— 5 you also, like living stones, are being built into a spiritual house[a] to be a holy priesthood, offering spiritual sacrifices acceptable to God through Jesus Christ." (1Peter2:4-5)
St.Irenaeus on Mary's role as Co-Redemptrix
As early as the year 200, the Church Father Irenaeus referred to Mary as the cause of our salvation (Latin: causa salutis) given her fiat ("let it be"). Theologians distinguish between "remote cooperation", by which she consents to the Incarnation and gives birth to the Son of God, and "immediate cooperation", in which she willingly unites herself to her Son's Passion and offers him back to the Father. Philosophers also draw a distinction between merit de condigno (Christ's merit), which is based on justice, and merit proprie de congruo (Mary's merit), founded on the friendship of charity.[ Garrigou-Lagrange, O. P., Reginald. Reality—A Synthesis Of Thomistic Thought, CH37: "Mariology" - Co-Redemptrix