The Stacks project

Lemma 99.4.2. In Situation 99.3.1. Let $T$ be an algebraic space over $S$. We have

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})}(T, \mathit{Isom}(\mathcal{F}, \mathcal{G})) = \{ (h, u) \mid h : T \to B, u : \mathcal{F}_ T \to \mathcal{G}_ T\text{ isomorphism}\} \]

where $\mathcal{F}_ T, \mathcal{G}_ T$ denote the pullbacks of $\mathcal{F}$ and $\mathcal{G}$ to the algebraic space $X \times _{B, h} T$.

Proof. Observe that the left and right hand side of the equality are subsets of the left and right hand side of the equality in Lemma 99.3.3. We omit the verification that these subsets correspond under the identification given in the proof of that lemma. $\square$

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