Yes, it is the water pump drive.

Yes, it needs to be tightened = there are two bolts, to loosen; (one is the pivot and other bolt is where the slot is.)

I tend to use a long, long pry bar, and once bolts are "slightly loose"; then push on the top of the pulley's bracket (above bolt with slot area) and that long  long prybar works good coming in from the very top of the engine = prybar is vertical, and just gives you a way to push straight downward. (you aren't using the prybar to "pry" anything)

Edit: this is on a semi-truck application, for MY procedure.

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(use longest bar.jpg)

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3126 water pump_in paint program.jpg (85KB - 445 downloads)

When everything in inside a truck = fan belt, fan shroud, fan, radiator, etc; you will find that square hole(for extension and ratchet) doesn't do you much good, because you are now trying to hold the 1/2" drive ratchet(one arm) and then stick your 2nd arm and 13mm wrench, in that tight space, and try to "snug-up" the bolt, that has the adjustment slot. {that photo is misleading to how much "stuff" is normally in a mechanic's way}

That's why I mentioned, earlier, "just slightly loose", on those idler bolts = then you can push down(or tap down), on a prybar against idler bracket, and the idler bracket will stay put (with a reasonable tight belt), and then you can finish tightening those 2 bolts.(with no ratchet in your way)

Note: That water pump belt must not take much effort to rotate, while engine is running, because I've seen many loose water pump belts, and they never complained of overheating. We would just go ahead and get the belt retightened, if we noticed it being way too loose.

That "belt tension" doesn't need to be super tight either, for that water pump = not like a fan and alternator belt application, where an engine fan really takes some HP, to drive it.