https://www.gamespot.com/articles/nintendo-switch-update-40-also-adds-gamecube-contr/1100-6454307/

Nintendo Switch's system update 4.0 is the gift that keeps giving. Released on October 21, the update introduced a number of key new features, including video capture support, a transfer data function, and wireless headphone support. Now it has been discovered that the update also enables GameCube controllers to be used with the console/handheld hybrid.

In order to use GameCube controllers with the Switch, you'll need the adapter sold for the Wii U around the time Super Super Bros. for Wii U was released. That can be plugged into the Switch dock, the controllers can be paired through the settings menu by pressing the L and R buttons. Once this is done, the controller is ready to use.