I am reading Olsson's book Algebraic Spaces and Stacks, and I am somewhat stuck in understanding one aspect of Example 1.2.4 (in page 10).
The goal is find a morphism of schemes $f: X\to Y$ which is quasi-compact and locally of finite presentation but not quasi-separated. Here is the construction. We let $Y =\operatorname{Spec} k[x_1, x_2, ...]$ denote the infinite affine space, and let $z$ be the closed point of $Y$ corresponding to the origin (where all $x_i=0$). Let $U=Y\setminus \{z\}$ is an open set of $Y$. It can be easily checked that $U$ is not quasi-compact.
Now, let $X$ be the scheme obtained by taking two copies of $Y$, and glueing them along the open set $U$. We can imagine $X$ to be an infinite affine space with "doubled" origin. Let's call $X_1, X_2\subset X$ the two subsets that we glued together (so $X_i\cong Y$). Consider the morphism $f: X\to Y$ which just restricts to the identity on each $X_i$. Then Olsson claims that $X_{1}\times_{Y} X_{2}\cong Y$ (which I kind of understand) and that the following diagram is a Cartesian diagram: $$ \require{AMScd} \begin{CD} U @>>> X_1\times_{Y} X_2\\ @VVV @VVV\\ X @>\Delta>> X\times_{Y} X \end{CD} $$ I understand that this gives the desired conclusion: indeed, if $f: X\to Y$ were quasi-separated, then the diagonal $X\to X\times_{Y} X$ would be quasi-compact, and hence its base-change $U\to X_{1}\times_{Y} X_{2}=Y$ would be quasi-compact, which is false. So my questions are:
1) How are we supposed to visualize the map $\Delta: X\to X\times_{Y} X$? If $X$ is the infinite affine space with doubled origin, what is the corresponding geometric picture for $X\times_{Y} X$? I am hoping this would clarify to me what $\Delta$ is.
2) Why is the above diagram cartesian?
Thanks!
Remark: There is already another post in MSE that shows $X$ is not quasi-separated as a scheme, i.e. the map $X\to \operatorname{Spec}(\mathbb{Z})$ is not quasi-separated. But my questions above still stand.