Global proof of deformation correspondence

Question

Given a scheme $X$ over a field $k$ and $Y$ a closed subscheme, it is well know that the first order deformations of $Y$ in $X$ correspond to the global sections of $\mathcal{N}_{Y/X}$ on $Y$. I know two proofs of this fact; the first reduces to the affine case, and the second works at the level of stalks using the appropriate ideal and structure sheaves. Even though the second proof is more "global" than the first, I am curious whether there is another proof of this correspondence without reducing to the affine case or working at the level of stalks.