Lemma 61.3.4. Let $A$ be a ring. Let $B \to C$ be an $A$-algebra homomorphism.
If $A \to B$ and $A \to C$ are local isomorphisms, then $B \to C$ is a local isomorphism.
If $A \to B$ and $A \to C$ identify local rings, then $B \to C$ identifies local rings.
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