  We discuss the folklore construction of the Gray tensor product of
2-categories as obtained by factoring the map from the funny tensor product to
the cartesian product. We show that this factorisation can be obtained without
using a concrete presentation of the Gray tensor product, but merely its
defining universal property, and use it to give another proof that the Gray
tensor product forms part of a symmetric monoidal structure. The main technical
tool is a method of producing new algebra structures over Lawvere 2-theories
from old ones via a factorisation system.
