Gluing morphisms of schemes

If $X={\bigcup }_i{U}_i$ is an open cover by open subschemes, and morphisms of schemes $f_i:U_i\to Y$ agree on every overlap, there is a unique morphism of schemes $f:X\to Y$ restricting to them. The underlying continuous maps glue because their restrictions agree. The maps of structure sheaves glue using the sheaf gluing axiom, and the required local condition on stalks can be checked on the original cover. No affineness or separation assumption on $X$, $Y$, or the overlaps is needed. This is the local construction behind affine test schemes determine scheme morphisms.