01 Window won't open
 

Hey everyone - the window on my Hyundai started to give me problems opening and closing. It's a 2001 5 speed SOHC 3 door hatchback with manual windows\locks, and I was wondering if someone could confirm for me whether this is the part I need to fix it:


It's been declining steadily in that it won't crank open without a lot of force sometimes, and now it just feels like it's jumping teeth. I'm not really sure what the apparatus is called but that kind of looks like something that would handle the window opening and closing. Anyone have any tips for me?
Thanks!