Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules on a scheme $(X,\mathcal{O}_X)$. I'm curious if the following equality holds for $U \subset X$ open and $P \in U$ (resp. $P \in \text{Specm}\ \mathcal{O}_X(U)$): $$\mathcal{F}(U)_P = \mathcal{F}_P$$ The LHS is the localization of the $\mathcal{O}_X(U)$-module $\mathcal{F}(U)$ at the maximal ideal $P$ and the RHS is the stalk at the point $P \in U$.
Maybe you need some assumptions on $\mathcal{F}$ or $(X,\mathcal{O}_X)$, feel free to add them in your answer. Can someone give me a rigorous proof of the above or link some reference? Many thanks in advance.