I have a Pfister widespread bathroom faucet.
As I was working on a leak and replacing the cartridge in one of the handles, I managed to loosen the other handle completely to where this part fell off:
It seems that it attaches to the bottom of the handle where the little metal pieces protrude as shown here from the top:
When I google, I find a lot of talk about using a basin wrench to tighten plastic nuts, but it seems like this contraption is something else entirely.
Can anyone enlighten me on (i) the names of the pieces and (ii) how I can re-attach the metal part from below the sink and then tighten it somehow?
UPDATE: Based on @JPhi1618's assistance below, you have to remove the handle (there is a cap which you can remove with your fingers or a flat-topped screwdriver, then a Phillips screw that you have to unscrew, THEN the plastic piece unscrews but for me this required quite a bunch of WD-40). Then, I made it to this:
Ended up buying two of these https://www.homedepot.com/p/RIDGID-10-in-Heavy-Duty-Straight-Pipe-Wrench-31010/100075882 RIDGID 10 in. Heavy-Duty Straight Pipe Wrenches. One is placed on flat portion of washer (?) shown by red arrow, other is placed on flat portion shown with yellow arrow.
Now, turn the wrench attached to the yellow arrow COUNTERCLOCKWISE while turning the washer shown with the red arrow CLOCKWISE. After some elbow grease, the washer started to turn, and I was then able to continue turning with my hand only. This lowered the carrier and I was able to re-attached the "gravity toggle" that fell off, then lower it under the sink, and turn the washer shown with the red arrow by hand COUNTERCLOCKWISE until it was tight.
Hope that helps someone! Still waiting for Pfister to tell me exact model of the faucet (I submitted a few tickets to them, haven't called yet).