Remark

When a species of structures $\Sigma$ has several principal base sets $x_1, \ldots, x_n$, and auxiliary base sets $\mathrm{A}_1, \ldots, \mathrm{A}_m$, a $\sigma$-morphism is a system $(f_1, \ldots, f_n)$, where $f_i$ is a mapping of $x_i$ into $y_i$ $(1 \leqslant i \leqslant n)$, and these systems of mappings satisfy conditions analogous to $(\mathrm{MO_{II}})$ and $(\mathrm{MO_{III}})$, which the reader may easily state for himself.