Lionel is holding a "sac à pof". It's a piece of fabric in which you put a mix of rosin and chalk or pure chalk, then close with a lace or cordelette. The loose side is used to hit the rock to put dust and chalk away, while the filled-in side is used to put chalk on your hands through the stitches of the fabric.

Lionel using the loose side to remove old chalk from the rock. It's less effective than the recycled toothbrush, but quicker and easier for higher holds.