existence of a factorization morphism related to extension fields Let $R$ be a Noetherian $k$-algebra, $k$ a field. Let $K$ be a field extension of $k$. Let $Q$ be a prime ideal of $R \otimes_k K$ such that $Q \cap R = p$. 

Question: What is an elegant way to see that the morphism $R \rightarrow (R \otimes_k K)_Q$ factors through $R_p \otimes_k K$?

Remark: In the context where this arises either $R$ is finitely generated as $k$-algebra or $K$ is a finite extension of $k$. I don't think though this affects the existence of the factorization in question.
PS: I am trying to show that $(R \otimes_k K)_Q$ is a localization of $R_p \otimes_k K$, but i can not formulate a rigorous argument.

In the category of $k$-algebras, $\_ \otimes_k \_$ is the coproduct. Thus giving a map $R_p \otimes_k K \to (R \otimes_k K)_Q$ is the same as giving maps $R_p \to (R \otimes_k K)_Q$ and $K \to (R \otimes_k K)_Q$ (here all maps are $k$-algebra maps), and there are such natural maps: $R_p \to (R \otimes_k K)_Q$ is just localization, and $K \to (R \otimes_k K)_Q$ is the composite $K \to R \otimes_k K \to (R \otimes_k K)_Q$. 
Note also that $R \to (R \otimes_k K)_Q$ factors through $R_p \to (R \otimes_k K)_Q$, which is the localization map given above.
