Let $f: X \longrightarrow S$ and $g: Y \longrightarrow S$ be two $S$-schemes. Then, one can form the fiber product $X \times_S Y$. My question is: Is there always a morphism of schemes $Y \longrightarrow_S X \times Y$ or $X \longrightarrow X \times_S Y$?
If we restrict ourselves to the affine case, say $X=Spec A$, $Y= Spec B$ and $S= Spec R$, then one can form a morphism from $A \longrightarrow A \otimes_R B$ obviously. Is gluing these affines change anything?
(Copied from my comment)
What you need is $A\otimes_RB\to A$, and the statement is false. Let $X,Y$ be two disjoint nonempty open subschemes of $S$. Then the fibered product is empty so there is no morphism into it.