Exercise 10, § 2

Let $\Sigma$ be the species of structures of a complete Archimedean ordered field. For each set A endowed with a structure of species $\Sigma$, let $\varphi_A$ be the unique isomorphism of A onto $\mathbf{R}$. If A, B are two sets endowed with structures of species $\Sigma$, show that the morphisms of A into B may be taken to be the mappings $f$ of A into B such that $\varphi_B(f(x)) \geq \varphi_A(x)$ for all $x\in A$. For this notion of morphisms show that there exist bijective morphisms which are not isomorphisms, although the species $\Sigma$ is univalent. \*