All outboards are basically the same and a water pump is routine maintnence. The star looking piece is the rubber water impeller. The vanes should be straight and soft. With age they take set and break. It spins in an aluminum or plastic housing and against a metal wear plate that is under my thumb. The housing and wear plate need to be inspected for groves but most just need an impeller.
The impeller and housing will burn in seconds if you run an outboard out of water.
The drive shaft should allways stay in the lower unit and then you pull the pump up off the shaft. On the folder the coupler is pressed on the end of the shaft so the drive shaft was pulled from the lower unit. Now my pinion gear is floating around in the gear housing. Part 18. Flashlight, screwdriver and patience got it together but the book has you disassembling the entire gear housing to get it together. I got lucky but once my impeller comes in I get to do it again.
They use a funky T shaped key or a round roll pin between the impeller and drive shaft. They are a pain because you have to flex the impeller vanes into the housing while keeping the pin in place. Grease on the pin, housing and impeller gets it together and then you must spin the driveshaft clockwise to make sure all the vanes are flopped in the right direction. Sometimes the vanes orient the right way when you spin it and sometimes you need to relocate them. If not it won't pump enough water. All outboards are like this and why they charge a couple hundred to do it.