How is the audit done?
You are the auditor, and you lead the audit, even if this is the supplier's premises. The supplier has been told that he shall spend one hour explaining the quality system and that you are not interested in knowing why you should buy his system during that hour.
The actual audit is when you ask for documentation to see what they do, and that they do what they claim. Ask questions starting with "Why", "How", "Show me" instead of questions that may be answered with yes or no. It is important that you follow the flow and understand what they are doing, rather than follow the checklists. Take a look at the checklist occasionally to see that you have covered the topics on the list. When you have seen what you want, the audit ends with a "wash-up meeting" where all findings are discussed. The findings shall be objective, i.e. they shall be proved as missing, erroneous, or inadequate according to the quality system or expectations, and acknowledged by both parties. A report is created after the meeting, and deviations shall be listed: What was discussed during the wash-up meeting, and nothing more.