Uniqueness of a local certain homomorphism (Etale Cohomology and the Weil Conjecture by Freitag, Kiehl)

Solution 1:

"First we reduce to the case where $B_1$ is actually etale $A$-algebra."

From the definition $$a_1: A\xrightarrow{} C_1\to S^{-1}C\simeq B_1$$ where $C_1$ is a etale A-algebra and $S$ a multiplicative closed subset of $C_1$.

Now from the remark (using quasi-finiteness and Zariski's main theorem: see paragraph after Def. 1.1 page 7) $C$ is actually a localization at $T$ of an etale $A$-algebra $D_1$ that is finite as $A$-module and $T$ a multiplicative closed subset. So we have a refined diagram

$$a_1: A\xrightarrow{\psi_1} D_1\to T^{-1}D_1 \simeq C_1\to S^{-1}C\simeq B_1.$$

Now apply the Nakayama argument to the composed maps as in your attempt $\phi_1, \phi_2: D_1\to B_2$, so we are done.