Group multiplication is a natural transformation between the appropriate functors. Similarly the defining operations of other algebraic structures are also nat. trans. between appropriate functors.
e.g. Let where sends to (the “squaring” functor), and is the forgetful functor. The set maps defined by the multiplication of are components of a natural transformation from to .
(ref. Adamek-Herrlick-Strecker, Abstract and Concrete Categories, 6.2 (2).)