1, having an affair with a customer of your company is a breach of contract, which is not only immoral but also illegal.
2. It is immoral to use the company's resources, such as telephone, internet, talents and related information, working hours, etc. Because the salary paid by the company includes the right to use these resources.
To put it bluntly, it is understandable to take a part-time job as long as it does not affect the current company and work, but it is hard to say as long as it has any impact on the current work.
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Seeing something added, I would like to add that this is a gray area and it is impossible to ban it, so how to do it depends on the company's norms and the professional ethics of consultants. I believe that most headhunters in China still have the minimum professional ethics, as long as they don't cross that line, they should all be ok. I'm afraid it's hard for headhunters with less than five years to have enough time and resources to do part-time jobs, while most headhunters with more than five years have no professional ethics.