The Iraqis do not have the luxury of conducting a siege: Prime Minister Haider al-Abadi has told his generals that dragging out the Mosul operation, now in its seventh month, would work only to the advantage of the Islamic State, which many in the West call ISIS or ISIL, but the Iraqis call Daesh.

This was not the mission that American military commanders envisioned for the counterterrorism service when they established it after the United States invasion in 2003. The force’s original mission was to conduct lightning raids against terrorists and insurgents.

“The CTS has made enormous sacrifices since 2014, and many of the old hand are dead, killed in Anbar Province and elsewhere,” said David M. Witty, a retired colonel with the United States Army Special Forces and former adviser to the counterterrorism service, known as CTS.

The militants’ basic strategy appears to be to focus much of their efforts on blunting the CTS attack, calculating that if they can stymie Iraq’s most experienced fighting force, the Iraqi government’s broader offensive will bog down. Now, the CTS is fighting on a southern front just to the west of Iraq’s federal police and other Interior Ministry troops.