Let $A$ and $B$ be subsets of abelian groups. A Freiman $2$-homomorphism is a function from $\phi:A\to B$ such that if $a_1 + a_2 = b_1 + b_2$, then $\phi(a_1)+\phi(a_2) = \phi(b_1) + \phi(b_2)$. If $\phi$ has an inverse that is also a Freiman $2$-homomorphism, then $\phi$ is said to be a $2$-isomorphism.
Now say $A$ contains $0$. Must $\phi(0) =0$ when $\phi$ is a $2$-isomorphism? My intuition says yes, but I have been unable to prove it. (We can't just say that since $0+a = a$, then $\phi(0) + \phi(a) = \phi(a)$, since we need another term on the right-hand side to apply the definition of $2$-homomorphism.)
If it is true, then does $\phi$ need to be a $2$-isomorphism? Or does $2$-homomorphism suffice? Thanks for the help!