Not all of the components are transferrable. The front derailer probably won't fit, the rear derailer probably will have a different hanger setup. The fork on the Mongoose may be a threaded fork and won't work on the carbon frame. Don't force any component to fit. Also, careful with the crank and BB. Ask if the BB and cups and cones will work in the new frame (Your current bike probably has the old cup and cone, square taper BB). Shifters will probably work and the wheels will work (if they're disc ready), but if your rear wheel is a non-quick release I would never put that on a carbon frame. It can be done, but I wouldn't. One thing to consider, the brakes on the new frame are probably disc only and your Mongoose probably has V or cantilever brakes. If you have canti brakes, you can't use the brake levers.
After writing all this, you need to post this on the MTB forum WITH A PIC OF THE MONGOOSE. It's very possible, nothing will be transferrable.