some of them email the offer letter.

but regardless, if the current co want to see your offer letter in order to justify increase your current pay, you mind as well leave for new co.

Because HR arent there to pay you fairly - they're there to ensure they pay you the minimum possible. That's why they needed the written offer to set a baseline and increase minimally or match. If your boss or HR values you, they'd offer what you're worth, not match what other companies think you're worth.

In one of my last company - I told my boss I am underpaid and on the spot they increased my pay. No third party offer or letters needed. They understand if they don't, I'll leave. BTW I left anyway....