I want to consider a module $P$ on a product $X \times Y$ of varieties over a field of characteristic zero, such that $P$ is flat over $X$. Furthermore I want to consider for each closed point $x$ on $X$ the natural map
$i_x:Y\rightarrow X\times Y$,
which comes from the cartesian diagram induced by the inclusion of $x$ in $X$, so naively it is just: sending a point $y$ on $Y$ to $(x,y)$ for that fixed $x$. With $P_x$ I want to denote the pullback of $P$ via $i_x$.
Assume that for every closed $x$ you have that $P_x$ is the skyscraper $k(y)$ for a closed point $y$ on $Y$. Then I can define a map of sets
between the closed points $Cl(X)$ and $Cl(Y)$ on $X$ and $Y$. My question: can you use this $f$ to define a real morphism between $X$ and $Y$ as varieties?
Remark: in the case I am interested in the f is bijective. Does this then imply that, if the extension exists, it is automatically an iso?