Travel Agent
The part you show is called a travel agent. While it does allow use of brake levers with less cable pull, the results can be disappointing. I have them on my touring bike, but the levers must be pulled nearly all the way back to the bar in order to get sufficient braking power with V-Brakes.
You can try them. They aren't too expensive, less than new levers and shifters, anyway.
Joferd