You can map available products on your store with private products by clicking the Override icon in My products or Mapping product button in My orders.

You will see Private products as an option when mapping.

Any product from your private product list can be selected and added to the list of alternate products.

After clicking Set, you will be directed to edit in Private product column and can start mapping as normal.

Remember to click Override after you finish mapping.

After mapping, the product, as well as any order includes that product (which has not been placed) will be updated with the new supplier.
The product details from storefront (such as title, description, images, variants, etc.) remain the same after changing suppliers.
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