In a 2-category like $\Prof$, the formulas $\tilde\vf : S T^o \cong U^oV$ with $T\dashv T^o, U\dashv U^o$ (or $\hat\vf : T_o S \cong VU_o$ with $T_o\dashv T, U_o\dashv U$) can be seen as exchange formulas between “2-fractions” or “2-co-fractions”, left and right.

In such a perspective of a 2-dimensional calculus of fractions, exact grids correspond to higher-order exchange formulas as $(\tilde\vf_1 U_2^oV_2)\circ (S_1T_1^o\tilde\vf_2) : S_1T_1^oS_2T_2^o \cong U_1^oV_1U_2^o V_2.$

Let $\mathbb K$ be a symmetric 2-category (cf. §1.4) and $X$ an object; we say that the diagram is an exact grid at $X$ if and only if the completed figure is a exact square (in the sense of condition 1·f [001V] or it opposite).

Similarly, a more complex grid will be called exact if completing it at its empty places with comma or cocomma squares we obtain a big exact square.

If $T : A\to X$ we say that a square $\vf_2 : US\To VR$ is $T$-exact if the grid is exact at $X$, and that $T$ is an opaque functor if the square is $T$-exact.

The characterisation of exact grids in $\Ab$ or $\Cat$ is immediate, thanks to the calculus of additive relations in $\Ab$ and the calculus of bimodules in $\Cat$. In a generic symmetric 2-category, we have to resort to an adequate `calculus of relations’, as described in the following section. In a category with a Yoneda structure, see the calculus outlined in §4.