Sheaves and Algebraic Geometry

For schemes $X,Y$, recall the natural map $$\alpha_{X,Y}\colon \operatorname{Mor}(X,Y) \to \operatorname{Hom}( \mathcal{O}_Y(Y), \mathcal{O}_X(X))$$ induced by taking global sections. ("Natural" meaning "functorial" in $X$ and $Y$.)