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

On the Uniqueness of L$_\infty$ bootstrap: Quasi-isomorphisms are Seiberg-Witten Maps