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e.g. a group is finite if there exists order : ℕ with isFinite : Bijection setoid (≡-setoid (Fin order)).
There's also the constructive notion of subfinite structures, which only have an injection to some finite set, for example arbitrary subgroups of finite groups.
I was imagining this going somewhere like {Relation.Binary,Algebra}.Finite.{Structures,Bundles}