Dictionary.com defines Adherence as a steady devotion, support, allegiance, or attachment. In short Adherence is when the person begin evangelized begins to take the faith into their own hands. The relationship that exists between them and Christ begins to become less dependent on the one who introduced them and they begin to pray and act on their own. This relationship that they are in begins to take control of more and more areas of their life. Their everyday decisions are affected by this relationships because they desire more and more to please God and to be in right relationship with Him.
Here are a few references from the Church to help better explain Adherence:
“To put it more precisely: within the whole process of evangelization, the aim of catechesis is to be the teaching and maturation stage, that is to say, the period in which the Christian, having accepted by faith the person of Jesus Christ as the one Lord and having given Him complete adherence by sincere conversion of heart, endeavors to know better this Jesus to whom he has entrusted himself: to know His “mystery,” the kingdom of God proclaimed by Him, the requirements and promises contained in His Gospel message, and the paths that He has laid down for anyone who wishes to follow Him.” – Catechesi Tradendae (20) – John Paul II
“The Christian faith is, above all, conversion to Jesus Christ, full and sincere adherence to his person and the decision to walk in his footsteps” – General Directory of Catechesis (53)
” Knowledge of the faith (fides quae) is required by adherence to the faith (fides qua).” – General Directory of Catechesis (85)