 Solution

A morphism of schemes is a morphism of locally ringed spaces. Thus it consists of a continuous map $f:|X|\to |Y|$ and a homomorphism of sheaves of ringssuch that, for every $x\in X$, the induced map on stalksis a local homomorphism: it carries the maximal ideal of the source into the maximal ideal of the target. Equivalently, the inverse image of the target maximal ideal is the source maximal ideal. The locality condition on every stalk distinguishes a morphism of schemes from a general morphism of ringed spaces.