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I'll start it off and maybe one of the more informed members can correct this reply: it is a fairly low pressure air pump and its there as part of the emissions system. As I recall it has a tube that pumps the air directly into the exhaust. For disconnecting, I doubt the motor would know the difference, unless it was the 02 sensor. But if you removed it, then the the fan belt wouldn't fit.
Tred99, shotgun posted about getting a proper belt fit after air pump removal. I'm not sure if it's applicable to your truck since you neglected to tell us what you have (OK, i'm going to take a wild guess that you do have some year of inline six....)
look:
https://www.ford-trucks.com/dcforum/DCForumID39/3133.html
In the future, if you find typing pertinent info tiresome you can just place the important stuff in your signature so as you post the site will do the rest for ya. It makes it nice in case one forgets to post make/model/year/options.
If you have to deal with emission testing leave it on -or draw a good diagram and keep all the parts in clean place. Sure you can disconnect the belt but you don't want the pump to sieze from non-use. If you have a O2 sensor that sends a signal back to the computer, you may need to leave it for the computer to run everything else correctly...not sure about the newer trucks. What year truck do you have?
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