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 S, U : \mathbf{Grp} \rightarrow \mathbf{Set} where S sends G \stackrel{f}{\longrightarrow} H to G^2 \stackrel{f^2}{\longrightarrow} H^2 (the “squaring” functor), and U is the forgetful functor. The set maps \tau_G : G^2 \rightarrow G defined by the multiplication of G are components of a natural transformation from S to U.
(ref. Adamek-Herrlick-Strecker, Abstract and Concrete Categories, 6.2 (2).)