A cartesian diagram? let $k$ be a field, and $X$ and $Y$ varieties over $k$. Let $L$ be an extension of $k$, and $X_L=X\times_k L$. Is the diagram
$$\require{AMScd} \begin{CD} X_L\times Y_L\times X_L @>>> X\times Y\times X\\ @VVV @VVV\\ X_L\times X_L @>>> X\times X \end{CD}$$
(all maps induced by/are projections)
a cartesian diagram? I didn't quite manage to prove the appropriate universal property. More generally, is there are sort of "base change" of cartesian diagrams which includes this?
 A: If by all the $\times$ you mean $\times_k$, then your diagram is not necessarily cartesian, as the example of $X = Y = \text{Spec}(\mathbb{R})$ and $L = \mathbb{C}$ shows (use that $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{C} \times \mathbb{C}$), so I assume the $\times$ on the right (resp. left) denotes fiber product over $k$ (resp. $L$). 
The easiest way to see these things is by using the limit preservation of the Yoneda embedding. So we need to see that evaluating your diagram on any test scheme $T$ yields a cartesian diagram of sets. Evaluating the upper left corner on $T$ gives
$$ \{ (x,l_1, y, l_2, x', l_3) \mid l_1 = l_2 = l_3 \} \subseteq X(T) \times L(T) \times Y(T) \times L(T) \times X(T) \times L(T),$$
(I'm lazy and use $L$ to denote $\text{Spec}(L)$) and evaluating on the lower left gives
$$ \{ (x,l_1,x', l_2) \mid l_1 = l_2  \} \subseteq (X(T) \times L(T))^2.$$
It follows that your diagram evaluated on $T$ can be identified with
$$\require{AMScd} \begin{CD} X(T) \times Y(T) \times X(T) \times L(T) @>>> X(T)\times Y(T)\times X(T)\\ @VVV @VVV\\ X(T) \times X(T) \times L(T) @>>> X(T)\times X(T) \end{CD},$$
which is visibly cartesian.
