Definition 2

Let E and E' be two sets each with a family of internal laws of composition $(x, y) \mapsto x *_{\alpha} y, \alpha \in I$. A mapping $f : E \to E'$ is a morphism if it is a morphism for each of the laws of composition $(x, y) \mapsto x *_{\alpha} y$.

Just as representations, so morphisms satisfy axioms (MO_I), (MO_{II}), (MO_{III}) of Set Theory, Chapter IV, § 2. Iff: $E \to E'$ is a morphism, $f(E)$ is a stable subset of $\mathbf{R}'$.