Let $L$ be a holomorphic line bundle and $L^*$ be the dual holomorphic line bundle.
So I believe what we want to show is that the following diagram commutes
(plus some condition on the restriction map on the fibers).
The "proof" I have seen just states to look the cocycles. I am not that familiar with cocycles, but I can make an attempt. Let $\phi_{ij}$ and $\psi_{ij }$ be the cocycles of $L$, $L^*$ respectfully and without proof I assume $\phi_{ij}\otimes \psi_{ij }$ is a cocycle of $L\otimes L^*$.
So $\phi_{ij}\otimes \psi_{ij }(x,v)\mapsto (x,g_{UV}\cdot g^*_{UV}(x)v)$
And this is where I get confused, Think somehow the transition maps $g_{UV}\cdot g^*_{UV}$ should give the identity but I not sure how it follows or if we should get the identity.
According to wikipedia we get that $g^*_{UV}$ is the inverse of the transpose of $g_{UV}$.