If $C^1, \ldots, C^k$ are members of a certain class of suitable categories (which contains those arising from models with dimension), $C = C^1 \times \cdots \times C^k, C'$ is a suitable category, $F: C \rightarrow C'$ is a partial recursive combinatorial functor satisfying a certain property (which, if $C = C^1$, is that $F$ is nonconstant) and $\mathscr{U} \in C$, then (1) if $F\mathscr{U}$ is regressive so is $\mathscr{U}$ as is each $\mathscr{U}^i$, and (2) if $F\mathscr{U}$ is Dedekind then each $\mathscr{U}^i$ is Dedekind.