Let $X$ and $Y$ be two schemes and suppose that $\{U_i\}_{i=1}^n$ is a cover by affines of $X$ and $\{V_i\}_{i=1}^n$ is a cover by affines of $Y$ and supposes there exists isomorphism $$f_i: U_i \to V_i$$
Since any projective scheme is glued out of its affine cover it seems that this should be enough to conclude that $X$ and $Y$ are isomorphic.
I find gluing constructions a bit hard to digest but it seems that the only way that $X$ and $Y$ would not be isomorphic is if there individual gluing functions,
$$ g_{ij}: U_{ij} \to U_{ij} $$ $$h_{ij}: V_{ij} \to V_{ij}$$
are somehow not compatible. How are $g_{ij}$ and $h_{ij}$ related if $X$ and $Y$ are isomorphic?