We had a customer try to back out of a contract by claiming the person signing didn't have authority. It didn't work because the person's manager (who has authority) was included in all of the communication.

Legally it didn't matter whether the signer had authority because the way the signer's company behaved during the signing process implied that the signer had authority.

E.g. If the CTO at a company tells a vendor to "send the contract over to my product manager" then the CTO created the impression with the counterparty that the product manager has authority, and the company will be hound to the contract based on that fact regardless of whether the product manager actually has authority or not.

I'm sure it's more nuanced than this, but my understanding is actual authority is less relevant than implied authority. E.g. if you have your board of directors take away the CEO's authority to sign a contract, it doesn't automatically invalidate everything the CEO signs, since a counterparty can reasonably assume that the CEO has authority just based on their job title.