Lemma 87.6.3. Let $A \to B$ and $B \to C$ be continuous homomorphisms of pre-adic rings. If $A \to C$ is adic, then $B \to C$ is adic.
Proof. Choose an ideal of definition $I$ of $A$. As $A \to C$ is adic, we see that $IC$ is an ideal of definition of $C$. As $B \to C$ is continuous, we can find an ideal of definition $J \subset B$ mapping into $IC$. As $A \to B$ is continuous the inverse image $I' \subset I$ of $J$ in $I$ is an ideal of definition of $A$ too. Hence $I'C \subset JC \subset IC$ is sandwiched between two ideals of definition, hence is an ideal of definition itself. $\square$
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