If we have two 2-functors

available, with the property that $J=J^o$ on objects, i.e. for all $X\in \mathbb{K}_o$ we have $JX=J^oX$, we can define a $J$-exact square as a square $\esq STUV$ such that $JS\circ J^oT\cong J^oU\circ JV$.

Conversely, let $\mathcal{E}$ be a class of squares in $\mathbb{K}$. We denote $\Span(\mathbb{K})$ the multiplicative system having as terms the spans $X \xot{a} R \xto{b} Y$ in $\mathbb{K}$, and where composition is defined up to isomorphism by a comma construction We denote $\equiv_\mathcal{E}$ the smallest equivalence relation on $\Span(\mathbb{K})$ such that $\Span(\mathbb{K})/{\equiv_\mathcal{E}}$ is a bicategory, and such that a span $(A,p,q)\equiv_\mathcal{E} (B, r,s)$ when there are 1-cells $u,v$ and 2-cells $\alpha : up\To vq$ and $\beta : ur \To vs$, both in $\mathcal{E}$. If we assume that $\mathcal{E}$ contains all the Yoneda squares, we recover the situation of the above construction.