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Let $L/K$ be a galois extension with $G:=\mathrm{Gal}(L/K)$, $X$ a $L$-scheme.

We have two definitions of the Weil restriction of $X$ :

1) If the contravariant functor $\mathrm{Res}^L_K(X) : (Sch/K) \rightarrow Set, T \mapsto X(T\times_K L) $ is representable, the corresponding $K$-scheme, again denoted by $\mathrm{Res}^L_K(X)$, is called the Weil restriction of $X$. [Néron Models, §7.6 (Bosch, Lütkebohmert, Raynaud)].

2) Let $W$ be a $K$-scheme with a map $p: W\times_K L \rightarrow X$ such that $W\times_K L$ is isomorphic to $ \prod\limits_{\sigma \in G} X^\sigma$ by the map $(p^\sigma)_{\sigma \in G}$, where $X^\sigma$ denoted the base change by $ \sigma^* : L \rightarrow L$ (the map induced by $\sigma$), then $W$ is called the Weil restriction of $X$.
[Adeles and algebraic groups, Weil, §1.3]

I would like to show that these two definitions define the same object. $\;\;\;\;\;$

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