Suppose $V$ is a holomorphic vector bundle on a Riemann surface X which is endowed with an anti-holomorphic involution $\sigma_X: X\longrightarrow X$. $V$ is said to have real structure if $\exists$ an anti-holomorphic isomorphism $\sigma_V: V \longrightarrow V$ (isomorphism as manifold) which is anti-linear on fiber, compatible with $\sigma_X$ and $\sigma_V \circ \sigma_V =id_V$ (For more details see section 3 of the paper "The moduli space of stable vector bundles over a real algebraic curve").
Suppose $V$ has real structure. Does this imply $V^*:= Hom(V,\mathcal{O}_X)$ has real structure?
Approach: For all $U$ open in $X$, define a sheaf theoretic map $$f(U): V^*(U) \longrightarrow V^*(U),\quad \phi_U \mapsto \phi_U \circ \sigma_V(U).$$ I can't able to show $f$ is anti-holomorphic isomorphism which is anti-linear on fiber. Is it going wrong somewhere?
