Remark 15.116.4. The construction in Lemma 15.116.3 satisfies the following “functoriality”. Suppose we have a commutative diagram
with injective horizontal arrows. Suppose given an element $f \in A'_1$ such that $(A'_1 \subset A_1, f)$ and $(A'_2 \subset A_2, f)$ satisfy properties (a), (b), (c) of Lemma 15.116.3. Let $n_{0, 1}$ and $n_{0, 2}$ be the integers found in the lemma for these two situations. Finally, let $B'_1 \to B'_2$ be a ring map, let $g \in B'_1$ be a nonzerodivisor on $B_1$ and $B_2$, let $n \geq \max (n_{0, 1}, n_{0, 2})$, and let a commutative diagram
be given whose horizontal arrows are isomorphisms and where $\varphi '_1(f) \equiv g$. Then we obtain commutative diagrams
where $(B'_1 \subset B_1, \varphi _1)$ and $(B'_2 \subset B_2, \varphi _2)$ are constructed as in the proof of Lemma 15.116.3. We omit the detailed verification.
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