  Let S be a distributive {&#8744;, 0}-semilattice. In a previous paper, the
second author proved the following result: Suppose that S is a lattice. Let K
be a lattice, let $\phi$: Con K $\to$ S be a {&#8744;, 0}-homomorphism. Then
$\phi$ is, up to isomorphism, of the form Conc f, for a lattice L and a lattice
homomorphism f : K $\to$ L. In the statement above, Conc K denotes as usual the
{&#8744;, 0}-semilattice of all &#64257;nitely generated congruences of K. We
prove here that this statement characterizes S being a lattice.
