In a group endomorphism between two sets, must the binary operation of the two sets be the same? Let $(G,\ast)$ and $(G,\cdot)$ be any two groups, where $\ast$ and $\cdot$ are distinct binary operations. Suppose $\phi:G\to G$ satisfies
$$
\phi(x)\cdot \phi(y)=\phi(x\ast y)
$$
for all $x,y\in G$. Is $\phi$ a group endomorphism? In other words, for a homomorphism $\phi:G\to H$ to be an endomorphism, do we simply require that the set $G$ equals $H$, or do we also require that the binary operations are equal as functions?

An endomorphism of $(G,*)$ is, by definition, a homomorphism from $(G,*)$ to $(G,*)$. Thus, in your formulation, we require $* = \cdot$ (as functions $G\times G \to G$).
