The Stacks project

Lemma 80.11.3. In Situation 80.10.6 the functor (80.11.0.1) is fully faithful on algebraic spaces separated over $X$. More precisely, it induces a bijection

\[ \mathop{\mathrm{Mor}}\nolimits _ X(X'_1, X'_2) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Spaces}(Y \to X, Z)}(F(X'_1), F(X'_2)) \]

whenever $X'_2 \to X$ is separated.

Proof. Since $X'_2 \to X$ is separated, the graph $i : X'_1 \to X'_1 \times _ X X'_2$ of a morphism $X'_1 \to X'_2$ over $X$ is a closed immersion, see Morphisms of Spaces, Lemma 66.4.6. Moreover a closed immersion $i : T \to X'_1 \times _ X X'_2$ is the graph of a morphism if and only if $\text{pr}_1 \circ i$ is an isomorphism. The same is true for

  1. the graph of a morphism $U \times _ X X'_1 \to U \times _ X X'_2$ over $U$,

  2. the graph of a morphism $V \times _ X X'_1 \to V \times _ X X'_2$ over $V$, and

  3. the graph of a morphism $Y \times _ X X'_1 \to Y \times _ X X'_2$ over $Y$.

Moreover, if morphisms as in (1), (2), (3) fit together to form a morphism in the category $\textit{Spaces}(Y \to X, Z)$, then these graphs fit together to give an object of $\textit{Spaces}(Y \times _ X (X'_1 \times _ X X'_2) \to X'_1 \times _ X X'_2, Z \times _ X (X'_1 \times _ X X'_2))$ whose triple of morphisms are closed immersions. The proof is finished by applying Lemmas 80.11.1 and 80.11.2. $\square$


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