Let $A \rightarrow B$ and $A \rightarrow C$ be commutative ring morphisms. I am trying to verify the universal property of the fibered coproduct for the ring $B \otimes_A C$ with maps $i_B: B \rightarrow B \otimes_A C$ and $i_C: C \rightarrow B \otimes_A C$ given by $b \mapsto b \otimes 1$ and $c \mapsto 1 \otimes c$ respectively.
So let $W$ be a ring such that we have ring morphisms $f_B: B \rightarrow W$ and $f_C: C \rightarrow W$. What I want to see is that there exists a unique map $f: B \otimes_A C \rightarrow W$ with the properties that $i_B \circ f = f_B $ and $i_C \circ f = f_C$.
Now to obtain $f$, the only way I can see is to apply the universal property of the tensor product. For this, I need an $A$-bilinear map $B \times C \rightarrow W$. The problem is that I don't see how to fabricate such a map from the maps $f_B: B \rightarrow W$ and $f_C: C \rightarrow W$. Rather, the setup allows me to find a map from the coproduct of $B$ and $C$, which at least in $Set$ is the disjoint union, so I don't see how that is useful.