 For one-sorted structures, it is very common to study different relations between them: homomorphisms, embeddings, isomorphisms, substructures and extensions (see Manzano 1989 [1999: 19–33]). One can define these relations for many-sorted structures in a similar vein. An homomorphism between two structures \(\mathcal{A}\) and \(\mathcal{A}^{\prime }\) with the same signature is a function \(\pi\) from the union of the universes of \(\mathcal{A}\) to the union of the universes of \(\mathcal{A}^{\prime }\) satisfying the following conditions: Firstly, the restriction of \(\pi\) to \(\mathbf{A}_{i}\) must be a function from \(\mathbf{A}_{i}\) to \(\mathbf{A}_{i}^{\prime }\), for each \(i\in Sort\). Secondly, if the n-tuple of elements \(\langle \mathbf{a}_{i},\dots ,\mathbf{a}_{n}\rangle\) is in the n-ary relation \(R^{\mathcal{A}}\) then \(\langle \pi (\mathbf{a}_{i}),\dots ,\pi (\mathbf{a}_{n})\rangle\) is in the relation \(R^{\mathcal{A}^{\prime }}\). Finally,