I am reading the proof of Nakano's Identity in Complex Geometry by Huybrechts. This is Lemma 5.2.3, for reference. If $\mathcal{A}_X^{p,q}$ denotes the sheaf of smooth $(p,q)-$forms on a compact Kähler manifold $X$, then the Hodge star operator $$ \star:\mathcal{A}^{p,q}_X\to\mathcal{A}_X^{n-p,n-q}$$ is defined using the pairing $\mathcal{A}^{p,q}_X\otimes \mathcal{A}^{n-p,n-q}_X\to \Bbb{C}$ coming from pairing to the volume form and integrating. If $\pi:E\to X$ is a Hermitian holomorphic vector bundle, we can define a new Hodge star $\star_E$ similarly by $$ \star_E:\mathcal{A}_X^{p,q}(E)\to \mathcal{A}_X^{n-p,n-q}(E^*)$$ by sending $\omega\otimes s\mapsto \star\omega\otimes s^*$ where $s^*$ is defined using the Hermitian metric.
Anyway, in Huybrechts' book - it is claimed that with respect to an orthonormal local trivialization of $E$ on an open $U$, $\star$ agrees with $\star_E$. I don't really see why this is true. Indeed, locally $E|_U\cong U\times \Bbb{C}^r$ so that a form in $\mathcal{A}^{p,q}_X(E|_U)$ is a vector valued differential form, rather than an "ordinary" differential form. Does he mean that $\star_E$ acts componentwise by $\star$? I am inclined to believe this, but where is the orthonormality assumption used here?