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Catechesis* [from the Greek καταχεω] is the act of handing on the Word of God intended to inform the People of God and candidates for initiation into the Church about the teachings of Christ, transmitted by the Apostles to the Church. Catechesis also involves the lifelong effort of forming people into witnesses to Christ and opening their hearts to the spiritual transformation given by the Holy Spirit.
A "catechism" is the name given to a written work that contains a summary the beliefs of the faith that is used as a teaching tool.
* From the USCCB