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How can one check that two line bundles on $\overline{M}_{0,n}$ coincide?

Let $X$ be the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$. Suppose I have two (big) line bundles $L$ and $L'$ on $X$ and that I want to show that they are the same element of $Pic(X)$. Of ...