I do not understand the following proof in the paper [Abelian varieties](http://van-der-geer.nl/%7Egerard/AV.pdf) by Edixhoven, van der Geert, and Moonen:


**(1.12) Rigidity Lemma.** Let $X$, $Y$ and $Z$ be algebraic varieties over a field $k$. Suppose that $X$ is complete. If $f: X \times Y \to Z$ is a morphism with the property that, for some $y \to Y(k)$, the fibre $X \times \{y\}$ is mapped to a point $z \in Z(k)$ then $f$ factors through the projection $\operatorname{pr}\_Y : X \times Y \to Y$.


Proof. We may assume that $k = \bar k$. Choose a point $x\_0 \in X(k)$, and define a morphism $g: Y \to Z$ by $ g(y) = f(x\_0,y)$ (here is the passage to to algebraic closed field involved, since otherwise we may not find such point $x\_0$…).


*Question:* Why $ k$ can be assumed to be algebraically closed? Assuming the lemma has been proved for fibre products $X\_{\bar{k}}$, $Y\_{\bar{k}}$, $Z\_{\bar{k}}$, how can we derive the statement for schemes over not algebraically closed $k$? It appears that there should be a little lovely diagram chase involved but I do not know how finally to construct the morphism $ Y \to Z$.


Finally there is the machinery of fppf-descent which justifies immediately the reduction to $k=\overline{k}$. But I would like to know if this can be also showed with elemenary methods — presumably a (tricky?) diagram chase. It looks rather similar to the problem [Proof of rigidity lemma](https://mathoverflow.net/questions/440404/proof-of-rigidity-lemma)
I posted recently, and there it turned out that the reduction to $k=\overline{k}$ can be justified by simple diagram chase. Is it here also possible to argue in similar way without ‘deep’ methods?



Reduction step to $k=\bar{k}$ in the proof of rigidity lemma
One way to argue is as follows: given a morphism $f \colon X \times Y \to Z$, consider its graph $\Gamma\_f \subseteq X \times Y \times Z$. Let $W$ be the scheme-theoretic image of $\Gamma\_f$ under the projection $\pi\_{Y \times Z} \colon X \times Y \times Z \to Y \times Z$, and let $X \times W \subseteq X \times Y \times Z$ be the preimage of $W$ under $\pi\_{Y \times Z}$.


Then we have $\Gamma\_f \subseteq X \times W$, and the statement that $f$ factors via $\pi\_Y \colon X \times Y \to Y$ is equivalent to the statement that $\Gamma\_f = X \times W$. Indeed, if $f$ factors through a morphism $g \colon Y \to Z$, then $W = \Gamma\_g$ and $\Gamma\_f = X \times W$. Conversely, if $\Gamma\_f = X \times W$, then the projection $\pi\_Y \colon W \to Y$ is an isomorphism since the same holds after multiplying with $X$ (this is a form of fppf descent, I suppose ― if you want you can choose a closed point in $X$ to reduce to the case of base change along a finite field extension). Thus $W$ is the graph of a morphism (namely $Y \underset{\pi\_Y}{\stackrel\sim\leftarrow} W \underset{\pi\_Z}\to Z$).


All of the above constructions are defined over $k$. Formation of scheme-theoretic image commutes with flat base change (on the level of sheaves, it is given by image factorisation $\mathcal O\_{Y \times Z} \twoheadrightarrow \mathcal O\_W \hookrightarrow \pi\_{Y \times Z,\*} \mathcal O\_{X \times Y \times Z}$), and checking whether an inclusion of subschemes is an equality can be done over $\bar k$. $\square$


