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The diverging views came after Fox News' Chris Wallace, who is moderating the third and final presidential debate, said it is not his job to judge the accuracy of the candidates' statements in real time.
"I do not believe that it's my job to be a truth squad. It's up to the other person to catch them on that," Wallace told Fox News earlier this month. "I certainly am going to try to maintain some reasonable semblance of equal time. If one of them is filibustering, I'm going to try to break in respectfully and give the other person a chance to talk."
Over the weekend, both of the nominees' campaigns managers, as well the executive director of the Commission on Presidential Debates, Janet Brown, weighed in on the role of moderators as fact-checkers.
Brown echoed Wallace's view, arguing that it's better for a moderator to facilitate and "for the candidates to basically correct each other as they see fit."
"I have to say, in our history, the moderators have found it appropriate to allow the candidates to be the ones who talk about the accuracy or the fairness of what the other candidate or candidates might have said," Brown told CNN in an interview on Sunday. "I think, personally, if you start to get into the fact-check, I am not sure — what is a big fact, and what is a little fact? And if you and I have different sources of information, does your source about the unemployment rate agree with my source?"
The Clinton campaign feels otherwise.
"All that we're asking is that if Donald Trump lies, that it's pointed out," Clinton campaign manager Robby Mook said Sunday on ABC News' "This Week."
"It's unfair to ask for Hillary both to play traffic cop with Trump, make sure that his lies are corrected, and also to present her vision for what she wants to do for the American people," he said.
Mook argued that fact-checking during the debate is necessary this election cycle.
"We normally go into a debate with two candidates who have a depth of experience, who have rolled out clear, concrete plans and who don't lie, frankly, as frequently as Donald Trump does," Mook continued. "So we're saying this is a special circumstance, a special debate, and Hillary should be given some time to actually talk about what she wants to do to make a difference in people's lives. She shouldn't have to spend the whole debate correcting the record."
Trump campaign manager Kellyanne Conway responded to the Clinton campaign's push for fact-checking during tonight's face off.
"I really don't appreciate campaigns thinking it is the job of the media to go and be these virtual fact-checkers and that these debate moderators should somehow do their bidding," Conway said in her interview on ABC News' "This Week."
Trump has also said the job of NBC's Lester Holt, who will be wrangling tonight's first debate, is to allow the candidates to correct each other.
"I think he has to be a moderator," Trump said on "Fox and Friends" on Thursday. "If she makes a mistake or I make a mistake, we'll take each other."
"I certainly don't think you want Candy Crowley again," Trump said of the CNN moderator who famously fact-checked Mitt Romney during the second 2012 presidential debate against President Barack Obama. "I really don't think that you want that. That was a pivotal moment in that debate, and it really threw the debate off, and it was unfair."
Trump continued, "I think you have to have somebody that just lets them argue it out."