Call a predicate $P$ defined on categories resplendent iff it satisfies the following condition: for all categories $\mathbf{D}$, if $P(\mathbf{D}),$ then for all categories $\mathbf{C}$, we have $P(\mathbf{D}^\mathbf{C}).$ Examples of resplendent predicates include: being terminal, being a setoid, being a preordered set, being a groupoid, being a truthvalue, being a category. Compare with the periodic table.
Question. Do other examples of resplendent properties exist?
Remark. Resplendency makes sense for predicates defined on the objects of any self-enriched category.
As long as we don't allow empty categories, the property "has at least $\kappa$-many objects" is resplendent for every cardinal $\kappa$.
Similarly, "has a non-well-orderable set of objects."