I would like prove that, under the conditions described below, no non-trivial variety exists.

Let $\mathcal{V}$ be a variety of algebras e.g. rings, semigroups, semilattices.

Further suppose that:

- The empty algebra exists i.e. $\mathcal{V}$ has no constants.
- The dual condition also holds. That is, let $\mathtt{1} \in \mathcal{V}$ be the one element terminal algebra. Then $id_\mathtt{1}$ is the only homomorphism whose domain is $\mathtt{1}$.

Any help much appreciated.