If you’re looking for ‘something you can only prove using category theory’, you’ll probably not be able to find much. If you’re looking for ‘something commonly proven using category theory, whose proof without invoking such theory is much less elegant and general’, you’ll find plenty in any area of mathematics that uses category theory.

There’s a whole section of Emily Riehl’s book Category Theory In Context on ‘theorems in category theory’ (since it’s famously said that there are none — you’re certainly not the first to level such an accusation!).