This passage opens with a direct quotation from CT ¶13 which discusses the Council of Trent (December 13th, 1545 to December 4th, 1563), a foundational council for the contemporary Catholic Church in light of the Protestant Reformation. As NewAdvent.org explains it:

Its main object was the definitive determination of the doctrines of the Church in answer to the heresies of the Protestants; a further object was the execution of a thorough reform of the inner life of the Church by removing the numerous abuses that had developed in it.

This ecumenical council is regarded as the epitome of the Catholic Counter-Reformation and gave rise to clear definition and reinvigoration in Church liturgy and practices, from naming the Vulgate (a late fourth-century Latin translation of the Bible) as the “official example” of Biblical canon to the introduction of the Tridentine Mass and the development of the Roman Catechism, which is the embodiment of the renewed catechesis of the time.

This paragraph also lists several contemporary Church figures who had a significant role (they’re saints, after all) in the definition and promulgation of this renewed catechesis.

Periods of renewal in the Church are carried on and cultivated by the successors of the apostles and their companions and accompanied by revitalized catechesis, particularly as depicted in the works of four of the Doctors of the Church listed herein: Saints Cyril of Jerusalem, John of Chrysostom, Ambrose, and Augustine. Each of these Early Church Fathers wrote definitive catechetical works which renewed the Church and its members, setting the foundations for the continuing catechesis the Church presents today.

Perhaps most familiar among these saints is Augustine of Hippo, whose works Confessions and City of God are widely read to this day. Check out Genius’s annotated Confessions here.

Cited for comparison is CT ¶12.

Catechesis, one of the elements of evangelization, which itself is responsible for “initiating the hearers into the fullness of Christian life,” is intertwined with the Church’s continuing existence both through growth in extent and number and the maturation of its members, refining their aspiration to fulfill God’s plan.

This paragraph quotes ¶13 of PJPII’s Catechesi Tradendae.

This section also quotes the 18th passage of PJPII’s Catechesi Tradendae, and serves to clarify that catechesis, as an element of evangelization, is also built upon various elements:
1. “the initial proclamation of the Gospel or missionary preaching to arouse faith” – to be preached to/taught concerning the “good news” of Jesus Christ
2. “examination of the reasons for belief” – to evaluate the purpose of belief
3. “experience of Christian living” – to initiate a “Christian life” in action and organization with elements such as prayer and attending mass
4. “celebration of the sacraments” – partaking in the sacraments, such as confession and communion
5. “integration into the ecclesial community” – integration into a parish/role in the Christian community
6. “apostolic and missionary witness” – to preach and evangelize, thus continuing catechesis as it began

This passage quotes CT ¶18, under the heading, “Catechesis as a Stage in Evangelization.” This passage seeks to clarify that “Catechesis cannot be dissociated from the Church’s pastoral and missionary activity as a whole” as well as “that there is no separation or opposition between catechesis and evangelization” – rather, evangelization is composed of many elements, and catechesis, instruction, is one of these elements which aids in “initiating the hearers into the fullness of Christian life.”

The text calls for comparison to one of Pope John Paul II’s post-synodal apostolic exhortations, Catechesi Tradendae, (subtitled “On Catechesis In Our Time”), which explores catechesis in the contemporary period, particularly in respect to the youth. As the first paragraph reads,

Very soon the name of catechesis was given to the whole of the efforts within the Church to make disciples, to help people to believe that Jesus is the Son of God, so that believing they might have life in His name, and to educate and instruct them in this life and thus build up the Body of Christ. The Church has not ceased to devote her energy to this task.