The following is example 6.1.2 from Fulton, Intersection Theory.
Denote projective coordinates on $\mathbb{P}^2$ by $[x,y,z]$, and on $Y=\mathbb{P}^2 \times \mathbb{P}^2$ by $([x,y,z],[u,v,w])$. Consider the lines $A=V(x) \subset \mathbb{P}^2$ and $B=V(z) \subset \mathbb{P}^2$. They meet in a point $P$. Define divisors $D_1=2A+B=V(x^2 z)$ and $D_2=A+2B=V(x z^2)$ in $\mathbb{P}^2$. Let $X$ be the closed subscheme $D_1 \times D_2 \subset \mathbb{P}^2 \times \mathbb{P}^2$, so $X=V(x^2 z,uw^2)$. Furthermore, let $f:V=\mathbb{P}^2 \to \mathbb{P}^2 \times \mathbb{P}^2$ be the diagonal.
Fulton claims that the intersection product $X \cdot V= 3\alpha + 3\beta + 3[P]$, where $\alpha,\beta$ are zero cycles of degree $1$ on $A$ resp. $B$.
I am trying to understand how this is calculated. I am aware that one could probably reverse the roles of $V$ and $X$ here to simplify things; I do not want to do that at the moment. First, one would calculate the intersection $W=V\times_Y X$. One sees that $W=V(x^2 z, xz^2) \subset \mathbb{P}^2$. If we denote $g:W \to X$, we need to calculate the cycle of $C=C_W V \subset g^* (N_X Y)$, and transfer it to $W$ via Gysin.
Because nothing interesting happens at infinity (or does it?), we can only look at the affine picture on $\mathbb{A}^2 =\{y\neq 0\}\subset \mathbb{P}^2$. The normal cone would then be $$\mathrm{Spec} \bigoplus_n (x^2z,xz^2)^n/(x^2z,xz^2)^{n+1}\simeq \mathrm{Spec}\ k[x,z,U,T]/(x^2z,xz^2,zT-xU),$$ whereas the pullback of the normal bundle would be $\mathrm{Spec} \bigoplus_n k[x,z]/(xz^2,x^2z)\otimes ((x^2z,uw^2)^n/(x^2z,uw^2)^{n+1})$.
How would one now calculate $[C]$, and in particular, the intersection class $s^*[C]$, where $s:W\to g^* N_X Y$ is the zero section?