Bundle-valued conjugate-linear Hodge star (${\ast }_E:A^{p,q}(E)\to A^{n-p,n-q}(E^{\ast })$)

For inner products linear in the first argument, define ${\ast }_E$ by contraction in $\psi \wedge {\ast }_E\eta =(\psi ,\eta )dV$. It is conjugate-linear and takes $(p,q)$-forms with values in $E$ to $(n-p,n-q)$-forms with values in $E^{\ast }$. With induced dual metrics, ${\ast }_{E^{\ast }}{\ast }_E=(-1{)}^{p+q}$.