  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 $\widetilde X$.
