Auto magma object

In mathematics, a magma object, can be defined in any category $\mathbf{𝐂}$ equipped with a distinguished bifunctor $\otimes :\mathbf{𝐂}\times \mathbf{𝐂}\to \mathbf{𝐂}$. Since Mag, the category of magmas, has cartesian products, we can therefore consider magma objects in the category Mag. These are called auto magma objects. There is a more direct definition: an auto magma object is a set $X$ together with a pair of binary operations $f,g:X\times X\to X$ satisfying $g(f(x,y),f(x^{\prime },y^{\prime }))=f(g(x,x^{\prime }),g(y,y^{\prime }))$ for all $x,x^{\prime },y,y^{\prime }$ in $X$. A medial magma is the special case where these operations are equal.