Exercise 2, § 2

Let S be a signed echelon type on $n + m$ letters (Exercise 1). Let $\Sigma$ be a species of structures having $x_1$, ..., $x_n$ as principal base sets and $\mathrm{A}_1$, ..., $\mathrm{A}_m$ as auxiliary base sets, whose typical characterization is of the form $s \in \mathfrak{P}(\mathrm{S}(x_1, \ldots, x_n, \mathrm{A}_1, \ldots, \mathrm{A}_m))$. Show that we may define a notion of $\sigma$-morphism for this species of structures as follows: given $n$ sets $\mathrm{E}_1, \ldots, \mathrm{E}_n$ endowed with a structure U of species $\Sigma$, $n$ sets $\mathrm{E}'_1, \ldots, \mathrm{E}'_n$ endowed with a structure $\mathrm{U}'$ of species $\Sigma$, and mappings $f_i : \mathrm{E}_i \to \mathrm{E}'_i$ $(1 \leqslant i \leqslant n)$, then $(f_1, \ldots, f_n)$ is said to be a $\sigma$-morphism if the mappings $f_i$ satisfy the following conditions :

(i) If S is a covariant echelon type, then

(ii) If S is a contravariant echelon type, then

Show that by assigning the variances suitably we may obtain in this way the definition of the morphisms for order structures, algebraic structures, and topological structures.