the state or process of becoming friends with someone or being befriended.
From befriend plus the '-ment' suffix (which creates nouns from verbs), following the pattern of words like 'agreement' and 'amazement.' Relatively rare in modern usage.
The '-ment' suffix is one of English's favorite ways to turn actions into abstract nouns—it lets us talk about qualities and conditions like 'befriendment' and 'bewilderment' as things we can discuss and study.
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