  We produce counterexamples to show that in the definition of the notion of
intertwining operator for modules for a vertex operator algebra, the commutator
formula cannot in general be used as a replacement axiom for the Jacobi
identity. We further give a sufficient condition for the commutator formula to
imply the Jacobi identity in this definition. Using these results we illuminate
the crucial role of the condition called the ``compatibility condition'' in the
construction of the tensor product module in vertex operator algebra theory, as
carried out in work of Huang and Lepowsky. In particular, we prove by means of
suitable counterexamples that the compatibility condition was indeed needed in
this theory.
