• Following the above remarks, to each “calculus of relations” on $\mathbb{K}$, we can associate a “notion of exactness” in $\mathbb{K}$, and conversely, if $\mathbf{E}$ is a class of distinguished squares n $\mathbb{K}$ that we deem “exact” we can build a calculus of relations on the bicategory $\Span(\mathbb{K})/{\equiv_\mathbf{E}}$.
• If $\mathbf{H}=\mathbf{H}(\mathbb{K})$ is the class of exact squares in the sense of condition 1·f [001V], we can show that

We know that conversely, $\mathbf{H}(\Ab)$ and $\mathbf{H}(\Cat)$ can be described in terms of $\Rel(\Ab)$ and $\Prof$.

• In theorem 3.4 of Meisen’s paper (Meisen, 1974), we can consider $\hom_{\mathbb{X}’}(X,Y)=\cate{Fix}(C_{X,Y})$, with $C_{X,Y}$ the triple induced on $\Span(X,Y)$ by the adjunction If $\mathbb{K}=\Ab$, we get $\mathbb{X}’=\Rel(\Ab)$. If $\mathbb{K}=\Cat$, we get $\mathbb{X}’=\Prof$.