I am new to algebraic geometry and category theory. I am wondering about the following functor is equivalence of categories or not.

Let $X$ be irreducible scheme and $x$ be its unique generic point. Let $\mathcal{O}_x$ be its local ring. Let $Coh(Spec~\mathcal{O}_x)$ be the category of coherent sheaves on $Spec~\mathcal{O}_x$. By definition $\mathcal{O}_x = \varinjlim_{x\in U}\mathcal{O}_X( U)$. The map $i:Spec~\mathcal{O}_x \to U$ is flat. Thus we have a natural functor $$i^* : \varinjlim_{x\in U} Coh(U) \to Coh(Spec~\mathcal{O}_x).$$

My question is, does $i^*$ is an equivalence of categories? If yes how to prove it?

I was trying to show it is fullu faithful and essentially surjective bu not able to do it!

Thanks in advance.