In the 2-category $\Cat$, an exact square (cf. (Guitart, 1980)) consists of four functors $S,T,U,V$ and a natural transformation $\varphi : U\circ S\To V\circ T$,

such that in $\Prof$ the 2-cell $\tilde\varphi : S\diamond T^o \To U^o\diamond V$ is invertible, or equivalently that the maps

induce an isomorphism

In addition to all examples in (Guitart, 1980) recalled in §3, we also recall that $\varphi$ is exact if and only if for all pointwise Kan extensions in $\Cat$ of the form the composite figure remains a pointwise Kan extension.

Bourn and Cordier have shown in (Bourn & Cordier, 1980) that the same preservation property of pointwise extensions is true in $\Prof$, from whence they deduce that an exact square is the most general situation where one has a factorization where $T_S$ and $T_V$ are, respectively, the codensity monads of $S$ and $V$.

We can also regard an exact square $\varphi$ as a generalized ternary decomposition of $B$, by virtue of the following example.

Let $E,D,M$ be three subcategories of $B$, and let $A$ be the category having as objects all morphisms of $D$ and as morphisms the commutative squares where $u,v$ are invertible. Then $(E,D,M)$ is a ternary decomposition of $B$ if and only if the square is exact, if we posit $\varphi=d$.

An exact square is said to be sub-exact (resp., sur-exact) if and only if for each pair $(x,y)$ of objects in $X\times Y$, $\tilde\varphi_{x,y}$ is injective (resp., surjective).

Any given square $\varphi$ can be decomposed into $\varphi = i \otimes_\epsilon s$, where $\epsilon$ is an exact square, $\otimes_\epsilon$ the multiplicative structure (cf. (Guitart, 1980), §1.5) induced by $\epsilon$, $i$ a sub-exact square, and $s$ a sur-exact square. This result is a 2-dimensional version of the 1-dimensional epi-mono factorization theorem.