Lemma 26.2.2. Let $X$, $Y$ be locally ringed spaces. If $f : X \to Y$ is an isomorphism of ringed spaces, then $f$ is an isomorphism of locally ringed spaces.

** An isomorphism of ringed spaces between locally ringed spaces is an isomorphism of locally ringed spaces. **

**Proof.**
This follows trivially from the corresponding fact in algebra: Suppose $A$, $B$ are local rings. Any isomorphism of rings $A \to B$ is a local ring homomorphism.
$\square$

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