Prop. 3.16 tells us that if $f: A \to B$ is a ring homomorphism and $\mathfrak p$ is a prime ideal of $A$ then $\mathfrak p$ is the contraction of a prime ideal of $B$ if and only if $\mathfrak p^{ec} = \mathfrak p$.
This is used in the proof of the Going-down theorem: If I understand correctly, AM claim it's enough to show that $B_{\mathfrak q_1} \mathfrak p_2 \cap A = \mathfrak p_2$.
Let $i: A \to B$ be the inclusion. Then $B_{\mathfrak q_1} \mathfrak p_2 \cap A = i^{-1} B_{\mathfrak q_1} \mathfrak p_2 = (B_{\mathfrak q_1} \mathfrak p_2 )^c$.
Let $f:B \to B_{\mathfrak q_1}$ be the (inclusion) map $b \mapsto \frac{b}{1}$.
Then $(B_{\mathfrak q_1} \mathfrak p_2 )^c = \mathfrak p_2^{ec}$.
Here is my question: In prop. 3.16, $()^e$ and $()^c$ are both with respect to the same homomorphism. What's the justification that we may use prop. 3.16 in the proof of the Going-down theorem where $()^e$ and $()^c$ are with respect to two different homomorphisms? Or am I misreading the proof of the Going-down theorem?