What is the restriction of a locally ringed space to an open subset U?

Question

Let $(X, \mathcal{O}_X)$ be a locally ringed space and $U\subset X$ an open subset.

For the definition of a scheme one considers $(U, \mathcal{O}_X|_U)$. What is $ \mathcal{O}_X|_U$ formally? Is it just the sheaf of all $\mathcal{O}_X(V)$ for all $V \subset U$ with the induced restriction maps?