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Let $X,Y,Z$ be smooth geometrically integral proper varieties over a field $K$ where $K/k$ is a finite extension of a number field $k$. Let $R|_{K/k}$ denote the Weil restriction. Suppose we have $K$-varieties morphisms $Y \to X$ and $Z \to X$ and let $Y \times_X Z$ be the fiber product.

Is $R|_{K/k}(Y \times_X Z)= R|_{K/k}(Y) \times_{R|_{K/k}(X)} R|_{K/k}(Z)$?

I know the above is true if $X= Spec K$ but I can't prove it in more generality. If it helps I can take $Y \to X$ and $Z \to X$ to be fppf but that's about it.

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    $\begingroup$ Weil restriction is a right adjoint, so commutes with projective limits, such as fiber products. Alternatively, you can check the relevant universal property directly. $\endgroup$ Commented Apr 1, 2016 at 23:07

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