How to set my default music library?

If i understand you correctly you want to use the music player that come preinstalled on your phone. If so no you can't have beats play that music which your devices personal library has. Vice versa you can't have Beats music play songs that are in your own personal library. Beats music has to play the music that you download to your library or stream since beats is a subscription music service. Kind of like you can have Netflix movies play through a movie app outside of Netfix since Netflix is a subscription music service nor can Netflix play movies you've downloaded or ripped from other sources. Hope this helps.