The Stacks project

Lemma 58.40.3. Let $X$, $Y$ be schemes. Any two morphisms $a, b : X \to Y$ of schemes for which there exists a $2$-isomorphism $(a_{small}, a_{small}^\sharp ) \cong (b_{small}, b_{small}^\sharp )$ in the $2$-category of ringed topoi are equal.

Proof. Let us argue this carefuly since it is a bit confusing. Let $t : a_{small}^{-1} \to b_{small}^{-1}$ be the $2$-isomorphism. Consider any open $V \subset Y$. Note that $h_ V$ is a subsheaf of the final sheaf $*$. Thus both $a_{small}^{-1}h_ V = h_{a^{-1}(V)}$ and $b_{small}^{-1}h_ V = h_{b^{-1}(V)}$ are subsheaves of the final sheaf. Thus the isomorphism

\[ t : a_{small}^{-1}h_ V = h_{a^{-1}(V)} \to b_{small}^{-1}h_ V = h_{b^{-1}(V)} \]

has to be the identity, and $a^{-1}(V) = b^{-1}(V)$. It follows that $a$ and $b$ are equal on underlying topological spaces. Next, take a section $f \in \mathcal{O}_ Y(V)$. This determines and is determined by a map of sheaves of sets $f : h_ V \to \mathcal{O}_ Y$. Pull this back and apply $t$ to get a commutative diagram

\[ \xymatrix{ h_{b^{-1}(V)} \ar@{=}[r] & b_{small}^{-1}h_ V \ar[d]^{b_{small}^{-1}(f)} & & a_{small}^{-1}h_ V \ar[d]^{a_{small}^{-1}(f)} \ar[ll]^ t & h_{a^{-1}(V)} \ar@{=}[l] \\ & b_{small}^{-1}\mathcal{O}_ Y \ar[rd]_{b^\sharp } & & a_{small}^{-1}\mathcal{O}_ Y \ar[ll]^ t \ar[ld]^{a^\sharp } \\ & & \mathcal{O}_ X } \]

where the triangle is commutative by definition of a $2$-isomorphism in Modules on Sites, Section 18.8. Above we have seen that the composition of the top horizontal arrows comes from the identity $a^{-1}(V) = b^{-1}(V)$. Thus the commutativity of the diagram tells us that $a_{small}^\sharp (f) = b_{small}^\sharp (f)$ in $\mathcal{O}_ X(a^{-1}(V)) = \mathcal{O}_ X(b^{-1}(V))$. Since this holds for every open $V$ and every $f \in \mathcal{O}_ Y(V)$ we conclude that $a = b$ as morphisms of schemes. $\square$


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