It means both as in both Hungarian the nationality and Hungarian the language?

Like in English: “The English speak English”.

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Yes, John 8:44 is very fitting:

You are of your father the devil, and your will is to do your father 's desires. He was a murderer from the beginning, and does not stand in the truth, because there is no truth in him. When he lies, he speaks out of his own character, for he is a liar and the father of lies.

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