Many calculi of relations in the literature are of the form $\Span(\mathbb{K})/R$ for some $\mathbb{K},R$ (cf. Monades involutives complémentées, pp. 35, 41 cited in (Guitart, 1977)). Similarly for the relations associated to the decompositions of Coppey (Coppey, 1971) on $\mathbb{K}$, the construction by Meisen (Meisen, 1974) of $\cate{Pull}(\mathbb{K})$ and $\cate{Rel}_M$ where $(E,M)$ is a Kelly decomposition, and the relations in a uniform Yoneda structure (§4).

Given a category $\mathbb{K}$, let $\cate{Rel}(\mathbb{K})$ denote the quotient of $\Span(\mathbb{K})$ by the relation $(R,a,b)\sim (R’, a’, b’)$ if and only if there are $u : R’ \leftrightarrows R: v$ such that $au=a’$, $bu=b’$, $a’v=a$ and $b’v=b$. The corresponding notion of exact square is that of a semi-cartesian square.

We can also obtain the semi-cartesian squares as the exact squares of a topos in terms of its usual calculus of relations. This is of course true for $\Set$, where condition 1·f [001V] has a different meaning.