= 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>.
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