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The phrase "a warm introduction" is correct and usable in written English.
It is typically used to refer to a friendly or cordial introduction between two people, often in a professional context.
Example: "I would appreciate it if you could facilitate a warm introduction to your colleague, as I believe we could collaborate effectively."
Alternatives: "a friendly introduction" or "a cordial introduction".
Exact(53)
Richard Pryor gives Newton a warm introduction.
Emily, thanks so much for such a warm introduction.
Get a warm introduction through a trusted contact.
And, after a warm introduction from a beaming Mr. Bloomberg, there was Mr. Koch himself.
Obey, a respected and important member of the house, gives Ferraro a warm introduction.
When it comes to meeting investors, I can't emphasize the importance of a warm introduction.
Similar(7)
Uniting their tastes and talents for the first time, the trio's debut release is a warming introduction to a group whose sound will make you dream of a vacation in a warmer place than you probably are reading this from.
And in time, regain the hope and confidence to flourish again," continued Harry, who received a fittingly warm introduction, complete with a military salute, from Gen. Powell.
LEONARD ANDAYA: Well, thank you very much, Kyla, for a very warm introduction.
He got an exceedingly warm introduction from long-time GOP activist Ralph Reed, whose Freedom and Faith Coalition organized the conference.
But there's a good chance you might not be connected to the right person from your alma mater that could make that warm introduction, and a recommendation for a plumber is probably always best from one of your friends.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com