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LingakuLingaku[verb] + 下来
把 + [noun] + [verb] + 下来
[adjective] + 下来
下来 after a verb often has nothing to do with coming down. Its extended senses share one picture: something goes from loose to fixed, from moving to still, or from the past into the present.
With verbs of recording or keeping – 写, 记, 拍, 录, 留, 存 – 下来 says the thing has been captured and stays.
停下来 "come to a stop" and 定下来 "be settled, decided" belong here too.
With 坚持, 活, 传 and 保留, 下来 says something has lasted from an earlier time to the present.
下去 is the mirror image: it carries something on from now into the future. 坚持下来 looks back on an effort completed; 坚持下去 urges keeping it up.
After an adjective, 下来 marks a change toward less: quieter, slower, darker. 起来 after an adjective marks a change toward more – 安静下来 against 热闹起来, 慢下来 against 快起来.
The object usually goes in front with 把, or as the topic. 下 without 来 lets it follow the verb: 请写下你的名字 "please write down your name".