A dagger 2-poset with tabulations, such as a tabular allegory.

Definition

A tabular dagger 2-poset is a dagger 2-poset$C$ such that for every object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A,B)$ there exist an object $\vert R \vert$ called a tabulation of $R$ and maps${p}_{A}f:\mathrm{Hom}(|R|,A)$ p_A:Hom(\vert f:Hom(\vert R \vert, A) and ${p}_{B}g:\mathrm{Hom}(|R|,B)$ p_B:Hom(\vert g:Hom(\vert R \vert, B) such that $R={p}_{B}g\circ {p}_{A}^{\u2020}{f}^{\u2020}$ R = p_B g \circ p_A^\dagger f^\dagger, and for every object $E:Ob(C)$ and maps $h:Hom(E,A)$ and $k:Hom(E,B)$, $k \circ h^\dagger \leq R$ if and only if there exists a unique map $j:E \to \vert R \vert$ such that $h = f \circ j$ and $k = g \circ j$.

Examples

The dagger 2-poset of sets and relations is a tabular dagger 2-poset.