On images of sofic systems

Let $\Sigma$ and $\bar\Sigma$ be finite alphabets. For topologically transitive sofic systems $ X\subset \Sigma^{\Bbb Z}$ and $\widetilde X\subset \widetilde\Sigma^{\Bbb Z}$ we give a necessary and sufficient condition for the existence of a homomorphism from $X$ to $\widetilde X$. For topologically mixing sofic systems $X \subset \Sigma^{\Bbb Z}$ and $\widetilde X\subset \widetilde\Sigma^{\Bbb Z}$, such that the topological entropy of $\widetilde X$ is less than the topological entropy of $X$, we give a necessary and sufficient condition for the existence of a homomorphism of $X$ onto $\widetilde X$.