Example 1

Take $\Sigma$ to be the species of order structures and let $\sigma\{x,y,s,t\}$ denote the set of all mappings $f$ of $x$ into $y$ such that the relation $(u,v)\in s$ implies $(f(u),f(v))\in t$. With the notation of Chapter III, § 1, this means that $u \leqslant v$ implies $f(u) \leqslant f(v)$, i.e., that $f$ is increasing. The verification of axioms $(\mathrm{MO_I})$, $(\mathrm{MO_{II}})$, and $(\mathrm{MO_{III}})$ is obvious.