dg.differential geometry - Definition of a complex structure on a vector bundle

For any antiholomorphic Diffeomorphism $fcolon Sto S$ we get a canonical identification $f^starbar K=K,$ $K$ and $bar K$ being the canonical and anticanonical bundle of the Riemann surface. A holomorphic structure on a complex vectorbundle $E$ is the same as an complex operator
$DcolonGamma(E)toGamma(bar KE)$ satisfying the (Cauchy Riemann) Leibnitz rule. (holomorphic sections are exactly the one in the kernel of D). (For higher dimensions this is not true.)

Now, $f^* E$ has a natural complex structure (it's just i).
Therefore one gets an anti-holomorphic structure $bar DcolonGamma(f^star E)toGamma(Kf^* E)$ satisfying the antiholomorphic Cauchy Riemann equation.
But the complex conjugate bundle $bar E$ also has a anti-holomorphic structure, since $overline{bar K E}=Kbar E.$ Therefore, $f^* bar E$ has a natural holomorphic structure.

These two holomorphic structures are not isomorphic in general:
In the case of a line bundle $L=E$ of degree $0$ one might see this as follows: every holomorphic structure $D$ gives rise to an unique unitary flat connection $nabla$ such that
$D=1/2(nabla+i*nabla).$ Then the anti-holomorphic structure on $bar L$ is given by $1/2(nabla-i*nabla)$ and, the unitary flat connection corresponding to the holomorphic structure on $f^* L$ is the connection $f^* nabla.$ But this connection is not gauge equivalent to $nabla$ in general: For example, on a square torus with f given by $zmapsto bar z$ the connection $d+c idx$ is not gauge equivalent to $d+ci dy$ for $cin Rsetminus 2pi Z.$

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