I may be overlooking something, but I'm not sure the original query and the suggested queries do the same thing. It looks like the original query wants to update the [supplier_name] column in [suppliers] if and only if there exists a customer in the [customers] table with a matching ID. The suggested queries would overwrite [supplier_name] to NULL if the condition did not exist, and the intention in the SQL statement that HughManity found elsewhere might have been to leave [supplier_name] alone in that case.
In other words, if we have customers and suppliers, and we manually assign them an ID and we make those ID's the same if the client is both a customer and a supplier of our company, then we want to "import" the customer name to the suppliers table, perhaps to fix up manual changes that other users have made, but leave everyone else's name alone.
Am I seeing that correctly? Or can someone give me a cyber knock alongside the head... ;-)