“Am I Being Scammed? Dealership Pressures Customer with Duplicate Credit Check for Approved Auto Loan”

Hi, I got an approved auto loan through a local federal credit union. They have provided me with a check for most of the cost of the car and the rest will come from me as a bank check. However, the car dealer says they still have to run my credit and submit the application to the same credit union where I got the loan. This sounds super shady. Am I being paranoid?