you are a coworker and have a conflict with another coworker who accuses you of being lazy, even though you started to work earlier than him. Is it your job to solve this problem?
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Speaking to your coworker directly is the first step. It’s called being an adult. It also makes it clear to the other person that you aren’t playing their games when you confront them directly. Of course, you also need to be clear that you hope to resolve things amicably. Then if that doesn’t make any difference, you go to management. Whether he’s wasting time isn’t your problem. Ignore it. But animosity toward you is your problem, and you’re the one solving it either way, whether directly or by going to management.