In Section 4 [001E], we define and characterise exact squares in a Yoneda structure in the sense of Street-Walters (Street & Walters, 1978), and we underline the possibility of defining pointwise squares in this framework. Maybe when we have sufficiently understood the link between exact and pointwise squares, we can construct a solid axiomatic approach to exactness in a 2-category.
Bibliography
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Street, R., & Walters, R. (1978). Yoneda structures on 2-categories. Journal of Algebra, 50(2), 350–379. https://doi.org/10.1016/0021-8693(78)90160-6Details