As the assessment phase ends, about a dozen teams of 12 U.S. military personnel each are embedding with the headquarters of brigade-sized and larger Iraqi units—outfits with thousands of troops. Their goal is to advise and assist the Iraqis on how to best battle ISIS. Assigning them to higher headquarters units is designed to keep the Americans away from the front lines and out of harm’s way.

“What we’re starting to see now, thanks in part to the assistance not only that the United States has given, but other countries, we’re starting to see the Iraqi security forces meld and form into much more capable fighting force than they were,” Rear Admiral John Kirby, the Pentagon spokesman, said Friday.

But as the number of troops increases, their roles will expand, Army General Martin Dempsey, chairman of the Joint Chief of Staff, told reporters in July. “We will match the resources we apply with the authorities and responsibilities that go with them based on the mission we undertake,” he said, “and that is to be determined.”