Refers specifically to the faith centered around the Pope and the Vatican, not all Christianity. He adheres to the Catholic tradition which is guided by the Vatican.
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Capitalization Matters
When referring to the church or its members, 'Catholic' is capitalized. Lowercase for general usage. The Catholic rituals are distinct from other Christian denominations.
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Cultural Nuance
In some contexts, 'Catholic' might imply adherence to specific traditions or values. As a Catholic family, they celebrate many traditional holidays.