The Stacks project

Lemma 61.3.7. Let $p : (Y, \mathcal{O}_ Y) \to (X, \mathcal{O}_ X)$ and $q : (Z, \mathcal{O}_ Z) \to (X, \mathcal{O}_ X)$ be morphisms of locally ringed spaces. If $\mathcal{O}_ Y = p^{-1}\mathcal{O}_ X$, then

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{LRS}/(X, \mathcal{O}_ X)}((Z, \mathcal{O}_ Z), (Y, \mathcal{O}_ Y)) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Top}/X}(Z, Y),\quad (f, f^\sharp ) \longmapsto f \]

is bijective. Here $\text{LRS}/(X, \mathcal{O}_ X)$ is the category of locally ringed spaces over $X$ and $\textit{Top}/X$ is the category of topological spaces over $X$.

Proof. This is immediate from the definitions. $\square$

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