Given an very ample line bundle $L$ on a curve we embed it into a projective space such that pullback of $\mathcal{O}(1)$ is $L$. Then we can identify the two. So in the category of vector bundles of the form $\bigoplus \mathcal{O}(n_i)$ where $n_i\geq 0$, obviously we can lift each object from the curve to the ambient projective space. The same is true for the morphisms, as the morphisms break down to direct sum of $\text{Hom}(\mathcal{O}(n),\mathcal{O}(m))=\Gamma(\mathcal{O}(m-n))$. My question: is there a functorial way to lift this category from the curve to the projective space? If so is it possible to make this functor exact at the neighborhood of the curve?