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 has a different meaning for bicategories, 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 $C$ such that
There is an object $0:Ob(C)$ such that for each object $A:Ob(C)$, there is a functional dagger monomorphism $i_{0,A}:Hom(0,A)$ such that for each object $B:Ob(C)$ with a functional dagger monomorphism $i_{B,A}:Hom(B,A)$, there is a functional dagger monomorphism $i_{0,B}:Hom(0,B)$.
For each object $A:Ob(C)$, $B:Ob(C)$, $E:Ob(C)$ with functional dagger monomorphisms $i_{B,A}:Hom(B,A)$, $i_{E,A}:Hom(E,A)$, there is an object $B \cup E:Ob(C)$ with functional dagger monomorphisms $i_{B \cup E,A}:Hom(B \cup E,A)$, $i_{B,B \cup E}:Hom(B,B \cup E)$, $i_{E,B \cup E}:Hom(E,B \cup E)$, such that for every object $D:Ob(C)$ with functional dagger monomorphisms $i_{D,A}:Hom(D,A)$ $i_{B,D}:Hom(B,D)$, $i_{E,D}:Hom(E,D)$, there is a functional dagger monomorphism $i_{B \cup E,D}:Hom(B \cup E,D)$.
The dagger 2-poset of sets and relations is an essentially algebraic dagger 2-poset.