\[\begin{array}{lcl} \tau(G){\rm Hom}(V,W)(v)\ni (\tau(g)\psi)(v) &=& g'\psi(g^{-1}v) \\ &=& g'(\lambda\otimes w)(g^{-1}v)\ (\exists (\lambda,w)\in V^*\times W,\ \psi=\lambda\otimes w\ at\ v) \\ &=& \lambda(g^{-1}v)g'w \\ \mu(G)(V^*\bigotimes W)(v)\ni (\mu(g)\lambda\otimes w)(v) &=& (g^*\lambda\otimes g'w) (v) \\ &=& \lambda(g^{-1}v)g'w \end{array}\]