For projects that have a single bidder on them and you want to move that bidder to another project, you can do so by pressing the "Move Company to the bidders list on an existing opportunity" button on the right side of an ITB (in the Project Inbox) or an Accepted opportunity on the Opportunities page.

Pressing this button will open a modal where you can enter the correct opportunity to move the bidder onto. This will move the bidder to the alternate opportunity and move the original opportunity to the project archive.
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If a project has multiple bidders attached already, the "Move company to the bidders list on an existing opportunity" button will not appear on the Opportunities page. You will have to go to the Project Details page to add or remove any bidders from the opportunity.

Related: Learn more about adding bidders to a project here.