  In the context of the recently proposed L$_\infty$ bootstrap approach, the
question arises whether the so constructed gauge theories are unique solutions
of the L$_\infty$ relations. Physically it is expected that two gauge theories
should be considered equivalent if they are related by a field redefinition
described by a Seiberg-Witten map. To clarify the consequences in the
L$_\infty$ framework, it is proven that Seiberg-Witten maps between physically
equivalent gauge theories correspond to quasi-isomorphisms of the underlying
L$_\infty$ algebras. The proof suggests an extension of the definition of a
Seiberg-Witten map to the closure conditions of two gauge transformations and
the dynamical equations of motion.
