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An essentially algebraic dagger 2-poset is a dagger 2-poset whose category of maps is a syntactic category for essentially algebraic theories?. “Cartesian” is a very overloaded term and “finitely complete” dagger 2-posets understood in category-theoretic terms are simply semiadditive dagger 2-posets.
An essentially algebraic dagger 2-poset is a dagger 2-poset such that
There is an object such that for each object , there is a functional dagger monomorphism such that for each object with a functional dagger monomorphism , there is a functional dagger monomorphism .
For each object , , with functional dagger monomorphisms , , there is an object with functional dagger monomorphisms , , , such that for every object with functional dagger monomorphisms , , there is a functional dagger monomorphism .
The dagger 2-poset of sets and relations is an essentially algebraic dagger 2-poset.