I think you're talking about using their "Partner Carrier" as one component of the FBA program
Track your shipment from your location all the way to the Amazon fulfillment center, and let the Amazon take care of the rest.
To select an Amazon-partnered carrier, simply:
You will then be given an estimate for the service and the opportunity to continue with an Amazon Partnered Carrier or to select another carrier."
Here's their overview of that part of the FBA program (Which answers you question about how to get it to their fulfillment center (warehouse)
Amazon-partnered carriers offer deeply discounted rates, and the cost is billed to your account as an "Inbound Transportation Charge." We will provide a shipping label you can print that you use with our carrier. The rates apply to deliveries within the 48 contiguous United States (not available for shipments originating in Alaska or Hawaii). You can cancel up to 24 hours after approving the charges by following these steps:
Important: Canceling the shipment does not cancel the Inbound Transportation Charge for the Amazon-partnered carrier. You must follow the steps above. If 24 hours have passed since approving the charges, you will not be able to cancel the charges or receive a refund.
After you've completed your shipment, give your box to your regularly scheduled UPS pick-up driver or take the package to a local UPS location.
Note: UPS has regularly scheduled web server maintenance every Sunday between 10 a.m. and 12 p.m. ET. During this maintenance period, UPS labels may not be available for FBA merchants using Amazon Partner Carrier shipments to fulfillment centers. If you encounter problems creating a UPS shipment with the Amazon Partner Carrier label during this time, try again after the maintenance period.
Amazon-preferred carriers regularly schedule deliveries to our fulfillment centers and are familiar with our receiving process. When you select an Amazon-Preferred Carrier, contact them directly to schedule a delivery and to get a price quote.
And here's the more general overview of the FBA program, generally, in case you haven't seen this:
Hope this helps
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