Lemma 77.24.1. Assume $B \to S$ and $(U, R, s, t, c)$ are as in Definition 77.20.1 (1). Let $\pi : \mathcal{S}_ U \to [U/R]$ be as in Lemma 77.20.2. Let $T$ be a scheme over $S$.

for every object $x$ of the fibre category $[U/R]_ T$ there exists an fppf covering $\{ f_ i : T_ i \to T\} _{i \in I}$ such that $f_ i^*x \cong \pi (u_ i)$ for some $u_ i \in U(T_ i)$,

the composition of the isomorphisms

\[ \pi (u_ i \circ \text{pr}_0) = \text{pr}_0^*\pi (u_ i) \cong \text{pr}_0^*f_ i^*x \cong \text{pr}_1^*f_ j^*x \cong \text{pr}_1^*\pi (u_ j) = \pi (u_ j \circ \text{pr}_1) \]are of the form $\pi (r_{ij})$ for certain morphisms $r_{ij} : T_ i \times _ T T_ j \to R$,

the system $(u_ i, r_{ij})$ forms a $[U/R]$-descent datum as defined above,

any $[U/R]$-descent datum $(u_ i, r_{ij})$ arises in this manner,

if $x$ corresponds to $(u_ i, r_{ij})$ as above, and $y \in \mathop{\mathrm{Ob}}\nolimits ([U/R]_ T)$ corresponds to $(u'_ i, r'_{ij})$ then there is a canonical bijection

\[ \mathop{\mathrm{Mor}}\nolimits _{[U/R]_ T}(x, y) \longleftrightarrow \left\{ \begin{matrix} \text{morphisms }(u_ i, r_{ij}) \to (u'_ i, r'_{ij}) \\ \text{of }[U/R]\text{-descent data} \end{matrix} \right\} \]this correspondence is compatible with refinements of fppf coverings.

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