Lemma 97.10.1. Let $S$ be a scheme. Let $Z \to B$ and $X \to B$ be morphisms of algebraic spaces over $S$. Then
$\mathit{Mor}_ B(Z, X)$ is a sheaf on $(\mathit{Sch}/S)_{fppf}$.
If $T$ is an algebraic space over $S$, then there is a canonical bijection
Proof. Let $T$ be an algebraic space over $S$. Let $\{ T_ i \to T\} $ be an fppf covering of $T$ (as in Topologies on Spaces, Section 73.7). Suppose that $(a_ i, b_ i) \in \mathit{Mor}_ B(Z, X)(T_ i)$ such that $(a_ i, b_ i)|_{T_ i \times _ T T_ j} = (a_ j, b_ j)|_{T_ i \times _ T T_ j}$ for all $i, j$. Then by Descent on Spaces, Lemma 74.7.2 there exists a unique morphism $a : T \to B$ such that $a_ i$ is the composition of $T_ i \to T$ and $a$. Then $\{ T_ i \times _{a_ i, B} Z \to T \times _{a, B} Z\} $ is an fppf covering too and the same lemma implies there exists a unique morphism $b : T \times _{a, B} Z \to T \times _{a, B} X$ such that $b_ i$ is the composition of $T_ i \times _{a_ i, B} Z \to T \times _{a, B} Z$ and $b$. Hence $(a, b) \in \mathit{Mor}_ B(Z, X)(T)$ restricts to $(a_ i, b_ i)$ over $T_ i$ for all $i$.
Note that the result of the preceding paragraph in particular implies (1).
Let $T$ be an algebraic space over $S$. In order to prove (2) we will construct mutually inverse maps between the displayed sets. In the following when we say “pair” we mean a pair $(a, b)$ fitting into (97.10.0.1).
Let $v : T \to \mathit{Mor}_ B(Z, X)$ be a natural transformation. Choose a scheme $U$ and a surjective étale morphism $p : U \to T$. Then $v(p) \in \mathit{Mor}_ B(Z, X)(U)$ corresponds to a pair $(a_ U, b_ U)$ over $U$. Let $R = U \times _ T U$ with projections $t, s : R \to U$. As $v$ is a transformation of functors we see that the pullbacks of $(a_ U, b_ U)$ by $s$ and $t$ agree. Hence, since $\{ U \to T\} $ is an fppf covering, we may apply the result of the first paragraph that deduce that there exists a unique pair $(a, b)$ over $T$.
Conversely, let $(a, b)$ be a pair over $T$. Let $U \to T$, $R = U \times _ T U$, and $t, s : R \to U$ be as above. Then the restriction $(a, b)|_ U$ gives rise to a transformation of functors $v : h_ U \to \mathit{Mor}_ B(Z, X)$ by the Yoneda lemma (Categories, Lemma 4.3.5). As the two pullbacks $s^*(a, b)|_ U$ and $t^*(a, b)|_ U$ are equal, we see that $v$ coequalizes the two maps $h_ t, h_ s : h_ R \to h_ U$. Since $T = U/R$ is the fppf quotient sheaf by Spaces, Lemma 65.9.1 and since $\mathit{Mor}_ B(Z, X)$ is an fppf sheaf by (1) we conclude that $v$ factors through a map $T \to \mathit{Mor}_ B(Z, X)$.
We omit the verification that the two constructions above are mutually inverse. $\square$
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