To allow someone to do something or to permit an action to happen.
From Old English 'lætan' meaning to allow, leave, or cause. It's one of the most ancient English verbs and is found in every Germanic language with similar forms.
The word 'let' is so old that it goes back before English split from German, and it literally means 'to leave someone to do something'—so when you 'let' your friend borrow something, you're 'leaving' it with them.
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