In Lack and Sobociński - Adhesive categories, Counterexample 7, the authors claim without proof that Cat is not an adhesive category. Seeing as Cat has all small (co)limits, the property of being an adhesive category that Cat doesn't satisfy must be that pushouts along monomorphisms are van Kampen squares. Note that the nlab article on extensive categories notes that Cat is extensive, so pushouts along coproduct injections are vK-squares.

My question is: Why are pushouts along monomorphisms not vK-squares for Cat? More explicitly, could somebody construct an example of small categories and functors such that one has a pushout along monomorphisms that is not a vK-square?

By Lemma 29 in the same paper, we should equivalently be able to consider Cat's satisfaction of the axioms of a High-Level Replacement Category equivalent to its satisfaction of the axioms of an adhesive category as Cat has an initial object. Cat certainly satisfies axioms 1, 2, and 4 by what's already been said. My suspicion is that it satisfies axioms 3 and 5 as well based on working out a basic example with diagram categories. So my guess, since Cat is not adhesive and thus also not a HLR category, is that Cat doesn't satisfy axiom 6. So alternatively, could somebody construct an example of small categories and functors such that one has a rectangle that is a pushout and a right square that is a pullback but a left square that is not a pushout?