Let $A,B, C$ be commutative rings. Let $i: A \rightarrow B$ and $j : A \rightarrow C$ be ring homomorphisms. Now, let $p$ be a prime ideal in $ B \otimes_A C$. Let $f$ be canonical map from $B$ to $ B \otimes_A C$ and $g$ be the canonical map from $C$ to $ B \otimes_A C$. Let $q$ be the inverse image of $p$ in $A$.
What is the relation between $ (B \otimes_A C)_p$ and $B_{f^{-1}(p)} \otimes_{A_q} C_{g^{-1}(p)}$? Is $ (B \otimes_A C)_p$ is isomorphic to $(B_{f^{-1}(p)} \otimes_{A_q} C_{g^{-1}(p)})_{p'}$? Here $p'$ is the inverse image of $p$ in $B_{f^{-1}(p)} \otimes_{A_q} C_{g^{-1}(p)}$ ( There is a canonical map from $B_{f^{-1}(p)} \otimes_{A_q} C_{g^{-1}(p)}$ to $ (B \otimes_A C)_p$).
I can show that there exists canonical maps using the universal property from $a: B \otimes_A C \rightarrow B_{f^{-1}(p)} \otimes_{A_q} C_{g^{-1}(p)}$ and $b: B_{f^{-1}(p)} \otimes_{A_q} C_{g^{-1}(p)} \rightarrow (B \otimes_A C)_p$ such that composition of $a$ and $b$ is canonical map from $ B \otimes_A C$ to $(B \otimes_A C)_p$. Using this I can show this is an injective map from $(B \otimes_A C)_p$ to $(B_{f^{-1}(p)} \otimes_{A_q} C_{g^{-1}(p)})_{p'}$. But I don't know how to prove it is surjective. ( I'm not even sure if that is true).