If it's not a big deal, why have people complained about US ground troops being assigned to a UN contingent in Europe during Clinton's term?
I just happen to think our Constitution states that our troops, ground, air or naval, should answer to our President and only our President, not some officer or leader of another country.
That's the difference between the UN and these NATO arrangements. In the UN deals, our soldiers are wearing blue helmets and are taking orders directly from foreign commanders. In these coalition deals, American soldiers are taking orders only from American commanders, and those commanders are participating with a group of other nations soldiers. It's not "under the command", it's "working alongside". Big difference.
The leader in charge of the task force is simply deciding the best course of action for the task force to take to accomplish its goals. If the US commanders don't want to participate in one of those actions, they can leave or bow out at any point they want. The coalition commander is basically saying "I think group x should go here, and group Y should go here". He's not giving orders, just direction to the group.