I am reading the notes on Inverse Image from professor Israel Vainsencher http://www.mat.ufmg.br/~israel/Ensino/Intersec/int3.pdf and having problems with the emxample $2$ on item $4.2$ page $21$: Let $X, Y$ be varieties over an algebraically closed field. Then we know that $X × Y$ is also a variety. Let $p : X × Y \rightarrow Y$ denote the projection. For each subvariety $V \subset Y$ we have $p^⋆[V ] = [X \times V]$. Let $r\in k(V)^*$ be a rational function and consider its image $p^*r$ in $k(X\times V)$. One checks easily that the cycles $p^*[r]$ and $[p^*r]$ are one and the same. It follows that $p^*$ induces a homomorphism
$p^*: A_k(Y)\rightarrow A_{k+n}(X × Y )$, where n = dim X.
My question is how to define the pullback of the rational function $p^*r$? I know that for being on varieties over an algebraically closed field, I have $k(X\times Y)\simeq Frac(k[X] \otimes k[Y])$, then I thought on defining $p^*:k(Y)\rightarrow k(X\times Y)$ by putting $p^*r=p^*(\dfrac{f}{g}):=\dfrac{f\otimes 1}{g\otimes 1}$, however I don't know how to pass from here to the definition of $p^*$ that is given to me on varieties: $p^*[V]=[X\times V]$.