inside the mouths of gift horses

I thought about this phrase which popped into my mind, wondering if I was mixing it up somehow as it didn't make sense:

"Don't look a gift horse in the mouth."

I was thinking about the Trojan horse; in that case it behooves you to look in the horse's mouth (unobtrusively, at least), to make sure there aren't enemy soldiers hiding inside it.

But this phrase apparently originated in regards to an actual living horse, not a wooden one, and the condition of its teeth:

https://www.phrases.org.uk/meanings/dont-look-a-gift-horse-in-the-mouth.html
https://en.wiktionary.org/wiki/don%27t_look_a_gift_horse_in_the_mouth