On continuity of homomorphisms between topological Clifford semigroups

Generalizing an old result of Bowman we prove that a homomorphism $f:X\to Y$ between topological Clifford semigroups is continuous if the idempotent band $E_X=\{x\in X:xx=x\}$ of $X$ is a $V$-semilattice; the topological Clifford semigroup $Y$ is ditopological; the restriction $f|E_X$ is continuous; for each subgroup $H\subset X$ the restriction $f|H$ is continuous.