After the no fly zone imposed in the 1990's in northern Iraq, sad dam gave permission for Turkish troops to enter Iraqi land to counter Kurdish insurgency.
To this day the Turkish military still has a camp in northern Iraq. However after the IS and the developments now the Turks are trying to train their own militia loyal to them to take over Mosul after IS is ousted.
However the Christian, Yezedi, Turkmen, Shabak, Armen, and Shias, and Sunni factions in the Pmf have made it clear, no one wants the MB to have control in the area because they are terrorists with another face. This step was taken after Barzani's visit to the gulf. It seems there was some sort of agreement between Erdogan, Barzani and the gulf states to create an Turkish proxy In Mosul. Once IS is ousted from Tel Afar, a force of no less then 10k PMF's will enter the area. The Turks better not let their proxies be in the way because as of right now, even the ethnic Kurdish yezedis want the peshmerga/ Turkish proxies out and instead want cooperation with the PMF of the Nineveh plains (mostly Shabak, Christian, Turkmen brigades backed by the southerners)