Any car experts that can help me figure this out? -- What is this thing that broke in my car?

(Car: Mini Cooper S, 2009)

I was replacing my cabin air filter and something extra fell out with the old filter. It seems like a valve of some sort, black with a green rubber edging.

I found a cabin air filter replacement video with an angle that shows the part: YouTube Video.

In the video, it becomes visible at the 0:35 mark when the old filter is pulled out, and again at the 0:54 mark right before the new filter is put in. It’s the part with the pale green outline.

In the video it looks like it’s closed on one opening. Directly “below” its position in the video is another opening of the same shape. It seems like it swings between these two openings, maybe to close one when the other is open? I’m not sure.

The valve has two plastic prongs that act as hinges (they sit inside circular holes at the top of the filter housing); one of the plastic prongs on mine is apparently broken, so it doesn’t stay in place.

I’m working out a way to replicate the prong on one side to hopefully get it working, but I’m wondering—

What is this part called, if it has its own name?

What is its function for the car?

How critical is its function?

Thanks!