[cat.theory]
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).)