I´m studying the construction of the moduli space of vector bundles and something came up to my attention. Suppose that we want to study the representability of the functor $$Vec_{X}:\{\text{k-schemes}\}\rightarrow\mathcal{Sets}$$ defined by $$Vec_{X}(S)=\{\text{vector bundles over } X\times S\}/\simeq$$ which assigns to each scheme $S$ the set of vector bundles over $X\times S$ up to isomorphisms.
Obviously the above functor is not representable because is not a sheaf due to the existence of non-trivial automorphisms. So, I have read that the sheafification of the above functor are the equivalence classes of vector bundles over $X\times S$ where two vector bundles $E, E'$ are equivalent if there exists a line bundle $L$ over $S$, such that $E\simeq E'\otimes p_{S}^{\ast}L$, where $p_{S}:X\times S\rightarrow S$ is the projection into the second factor. I don´t see how to prove that the sheafification gives this equivalence relation. This would help me to understand better other functors such the Pic functor. Thank you very much for your time.


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