Auto magma object

In mathematics, a magma object, can be defined in any category $\mathbf {C}$ equipped with a distinguished bifunctor ${\displaystyle \otimes :\mathbf {C} \times \mathbf {C} \rightarrow \mathbf {C} }$. Since Mag, the category of magmas, has cartesian products, ergo we can consider magma objects in the category Mag. These are called auto magma objects. There's a more direct definition: an auto magma object is a set $X$ together with a pair of 2-place operations ${\displaystyle f,g:X\times X\rightarrow X}$ satisfying ${\displaystyle g(f(x,y),f(x',y'))=f(g(x,x'),g(y,y'))}$. A medial magma is the special case where these operations are equal.