Dual of bundle on the language of projective modules (via Serre-Swan)

Question

Let $E$ vector bundle on some space $X$. Let $\overline{\mathbb{C}}$ is one-dimensional complex conjugate vector space. I am trying to understand why $E^*$ is corresponds to $\operatorname{hom}_{C(X)}(\Gamma(X,E),\Gamma(X,X \times \overline{\mathbb{C}}))$. It is not completely obvious to me even how $\Gamma(X,X \times \overline{\mathbb{C}})$ can be realized as direct summand of some free module $C(X)^n$ (via Serre-Swan $\Gamma(X,X \times \overline{\mathbb{C}})$ must be projective finitely generated $C(X)$-module, right?). I will be grateful for any tips!