Let $f\colon X\to Y$ be a morphism between irreducible $\mathbb{C}$-varieties, such that $f$ is 1-1 and surjective, but $df$ fails to be injective along a subvariety $Z$. (This forces $Y$ to be singular). For example one can consider $X=\mathbb{C}^2$ with coordinate $x,y$, and $Y\subset \mathbb{C}^4$ is parametrized by $(x^4, x^2, x^3, y)$, defined by $f$. Take $X\to X\times_{\mathbb{C}} X$ and $Y\to Y\times_{\mathbb{C}} Y$ to be diagonal embedding, and $X\to X\times_{\mathbb{C}} Y$ to be the graph embedding of $f$.
My question is, in this case, with the (possible non-reduced) scheme structure,
Is the fiber product $Y\times_{Y\times Y} X\times Y$ just $X$ ?
What is the fiber product $Y\times_{Y\times Y} X\times X$ ?
