We explain how, in the context of a semi-abelian category, the concept of an internal crossed square may be used to set up an intrinsic approach to the BrownLoday non-abelian tensor product.
Van der Linden, T., & Di Micco, D. (2020). An intrinsic approach to the non-abelian tensor product via internal crossed squares. Theory and Applications of Categories, 35(34), 1268-1311. https://hdl.handle.net/2078.5/121317 (Original work published 2020)