Exact squares in $\Cat$, in representable 2-categories or in 2-categories equipped with a Yoneda structure have been introduced in (Guitart, 1980) and, subsequently, the theory has been developed by Van den Bril (Van den Bril, 1980), (Van den Bril, 1982) Guitart (Guitart, 1982), Guitart-Lair (Guitart & Lair, 1980) (in the case of $\Cat$, see here some complementary remarks to §12).

One first reason for the success of these 2-dimensional exact squares is that more or less everything one can do in $\Cat$ (e.g., defining comma and co-comma squares, rich functors, opaque functors, co-fully faithful functors, limits and absolute extensions, adjunctions and relative adjunctions, pointwise Kan extensions) is naturally and directly expressible in terms of exact squares.

A second reason is that, in the case of $\Cat$, profunctor composition seen as bi-fibrations is performed again taking into account the key observation of our §2.