Lemma 25.3.5. Let $f : X \to Y$ be a morphism of locally ringed spaces. Let $U \subset X$, and $V \subset Y$ be open subsets. Suppose that $f(U) \subset V$. There exists a unique morphism of locally ringed spaces $f|_ U : U \to V$ such that the following diagram is a commutative square of locally ringed spaces

\[ \xymatrix{ U \ar[d]_{f|_ U} \ar[r] & X \ar[d]^ f \\ V \ar[r] & Y } \]

**Proof.**
Omitted.
$\square$

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