Primitively recursive categoricity for unars and equivalence structures

This continues the study of the primitively recursive categoricity of structures for which there exists a primitively recursive decision algorithm with witnesses of all $\Sigma$-formulas. Considering the equivalence structures, we find a complete criterion for primitively recursive categoricity over the class $K_\Sigma$, which coincides with the already known criterion for computable categoricity. As regards unars, the structures with one arbitrary unary function, we distinguish some conditions for primitively recursive categoricity over $K_\Sigma$ and also for the absence of this categoricity. In particular, we find a full description of primitively recursive injective unars categorical over $K_\Sigma$.