Sheaves and Algebraic Geometry

Recall that in the definition of a morphism of schemes (or of locally ringed spaces) $f\colon X\to Y$, the structure map $f^\sharp\colon \mathcal{O}_Y \to f_\ast \mathcal{O}_X$ induces a stalk map $f^\sharp_x\colon \mathcal{O}_{Y,f(x)} \to \mathcal{O}_{X,x}$ at every point $x\in X$.