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    $\begingroup$ I don't think the bilinear map is an important part of the setup. What you want is a wedge product $\Omega^*(M,E) \otimes \Omega^*(M,F) \to \Omega^*(M, E \otimes F)$. Then you can simply apply $\mu$, because a linear bundle map E -> E' induces a map from E-valued forms to E'-valued forms. Phrased like this I believe there's a discussion of bundle-valued wedge products in Sharpe's differential geometry textbook, but I don't have access to a copy to check right now. $\endgroup$ Commented Jul 31, 2021 at 16:15