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to become attached (to); to take (to); to become affectionate (with); to be tamed; to get close (e.g. to someone emotionally); to become intimate (with)
「懐く」 has another reading with different meanings.
近所の野良猫がすっかり懐いてくれました。
The neighborhood stray cat became attached to me.
子供たちは新しい先生にすぐ懐いています。
The children are attached to their teacher.
犬は飼い主に懐くのがとても早いです。
Dogs quickly become attached to their owners.