Debate moderators have an obligation to fact check the candidates. If one of them is making false statements, they should be challenged and made to answer questions. Otherwise, candidates will be able to say false things in order to present their issues and themselves in the best possible light. This keeps candidates honest.
No, Debate Moderators should not just stick to moderating. Moderators are usually highly educated and/or extremely well-informed and accomplished individuals and should be able to call out when someone is speaking untruths. Donald Trump is the king of hyperbole, and loves to just make things up as he goes depending on the temperature of his audience, a tactic that will be a hindrance to him in an intelligent debate.
Moderators have a duty to check facts. Presidential debates are not about who has the best arguments like in a high school or college debate. They are about who is the best presidential candidate and who should win the race. Fact checking is an integral part of that, and the moderators have a duty to uphold that.
Moderating involves directing the conversation. If someone is blatantly lying in a debate, then the moderator should call out that person. It is not fair to send incorrect information to the viewing public. Also, it is unfair to the other candidate who likely prepared their arguments based on real facts instead of invented beliefs. Anyone can win an argument if they are allowed to make up information to back up their point. Moderators should definitely call out lies to make sure the debate is fair.