We have the following automata $A$:
My task is to find automata $B$ and $C$ such that both of them admit a homomorphism to $A$, but $B$ is not homomorphic to $C$ and vice versa.
So far, we know that both $B$ and $C$ have to be automata accepting the same language as $A$. Furhermore, it is clear that it is language of binary words with the even number of zeroes.
My definition of automata homomorphism is the same as in here.