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A morphism of ringed spaces $f:(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ consists of a continuous map $f:X\to Y$ and a morphism of sheaves of rings $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$.

A ringed space $(X,{\mathcal{O}}_X)$ is a topological space $X$ equipped with a sheaf of rings ${\mathcal{O}}_X$. A morphism consists of a continuous map $f:X\to Y$ and a compatible morphism $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$.