Let $X$ be a compact complex manifold, $\sigma:\hat{X}\to X$ is the blow up of a point $x\in X$. Let $E:=\sigma^{-1}(x)$ and $L\to X$ be a line bundle, then how to give a rigorous proof to show that
$\sigma^*L\otimes\mathcal{O}_E\cong L(x)\otimes\mathcal{O}_E$?
I can see $\sigma^*L\otimes\mathcal{O}_E$ is the pullback of $L(x)$ along $E\to\{x\}$, but I don't use how to put this fiber product into the tensor product of the sheaves. Can you illustrate the proof?