Ok so you have climbed you first 3 pitch route, set the anchors and put your mate on belay. And he is off. Move after move he’s getting more pumped and eventually he falls off, unable to continue the route.
Rescuing is pretty straight forward using assisted and unassisted hoist. It’s pretty simple start off with taking in all the slack (if any) and tying off your belay.
Belay will play its role in hoisting up. Now attach a French prussic to the live rope (one going to/from the climber) and clip it to your belay loop. This will act as a brake and will let the rope move only one way.
Now take your dead rope and simply clip a screwgate on it and lower it to the climber. Second climber needs to clip this on to his belay loop.
Take in all the slack. Now you can untie the belay watch for a little bit of slack this will introduce to your system. Now the second will pull on to rope that travels to him from the top and you need to pull on rope coming back from the second.