Morphism of locally ringed spaces ($f_x^{\#}:{\mathcal{O}}_{Y,f(x)}\to {\mathcal{O}}_{X,x}$)

A morphism is a continuous map $f:X\to Y$ together with a homomorphism ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$ of sheaves of rings whose induced map on every stalk is a local homomorphism. Applied to schemes, this is exactly a morphism of schemes.