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,

1. Is the fiber product $Y\times_{Y\times Y} X\times Y$ just $X$ ?

2. What is the scheme structure of the fiber product $Y\times_{Y\times Y} X\times X=X\times_Y X$ ?

Thanks for the comment abx. Probably I should be more precise, what I would like to ask is, what is the scheme structure of $X\times_Y X$? For example, the base set is just the diagonal $\Delta_X$ union with the double point set, but as a scheme there should be a non-reduced structure along the non-immersion locus, I suppose. My question was to ask this scheme structure.

Let me explain what I thought. My initial example was, for smooth $X$ and non-smooth$Y$ (both n-dimensional, $f$ is 1-1, surjective with non-immersion locus $Z$), we have the following equation ($h\colon X\times_Y X\to Y$) $$h_*s(X\times_Y X, X\times X)=c(TY)\cap s(Y, Y\times Y) $$ Now assume that the double point set is $0$ dimensional, then $X\times_Y X$ is disjoint union of two schemes, thus $C_{X\times_Y X} X\times X$ has components over the isolated points and over the diagonal. Thus $$s(X\times_Y X, X\times X)=A+ \sum_{\text{P double point}} [P] $$ where $A$ is a class in $\Delta_X$. My initial point was to dig what is $A$.