Let be a Lawvere theory with generic object . For , the full subcategory of generated by the cartesian powers of is also a Lawvere theory, which we denote by . In the case of an annular theory (the theory of modules over a ring that we also call ), this is the construction of matrices over . If we denote by the application of this construction to the initial theory (the theory of sets), then we may identify with the tensor product theory .
It is an amusing exercise to present in terms of generating operations and relations between them.